Puzzle 1092 Consecutive integers of the same parity

Problems & Puzzles: Puzzles

Puzzle 1092 Consecutive integers of the same parity

JM Bergot sent the following nice puzzle:

33=3*11 and 33+3+11=47

35=5*7 and 35+5+7=47

329=7*47 and 329+7+47=383

341=11*31 and 341+11+31=383

Q1. Can you find a larger pair giving the same

prime result?
Q2. Is it possible for more than

two to give the same prime result?

During the week 25.30 June 2022, contributions came from Paul Cleary, Emmanuel Vantieghem, Jean-Marc Rebert, Gennady Gusev, Simon Cavegn, Giorgos Kalogeropoulos

***

Simon wrote:

I noticed the sum results that have many solutions are often one less than numbers in https://oeis.org/A015702/b015702.txt
To find a first example I used: 2^8 * 3^5 * 5^3 * 7^2 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 - 1 = 13464187271799263999 prime.

Here the first and last few lines. For all 2623 lines, see attached Puzzle1092_Q2.txt

10098140453849447997=3*3366046817949815999 and 10098140453849447997+3+3366046817949815999=13464187271799263999
12502459609527887987=13*961727662271375999 and 12502459609527887987+13+961727662271375999=13464187271799263999
12903179468807627977=23*561007802991635999 and 12903179468807627977+23+561007802991635999=13464187271799263999
..
13464187264460281021=3638178179*3700804799 and 13464187264460281021+3638178179+3700804799=13464187271799263999
13464187264460341061=3641889599*3697033339 and 13464187264460341061+3641889599+3697033339=13464187271799263999
13464187264460545981=3666482819*3672235199 and 13464187264460545981+3666482819+3672235199=13464187271799263999

An even larger solution for both Q1 and Q2: (file expires in 6 days): https://filebin.net/zfc7f3b8i3bjqt9h

***

Emmanuel wrote:

Looking at the examples given by Bergot, I restricted myself to odd semiprimes
(if even semiprimes would be involved, I think Bergot would have started with the pair {14, 15}, producing 23).

So, let m = p q be a semiprime, p, q being odd primes. Put f(m) = m + p + q.
I examined all semiprimes below 200000000.
Then, the smallest set of two semiprimes that have the same prime f-value is indeed {33, 35}.
The biggest pair I found is {179999637,199999595}, producing the prime 239999519.
There are 765267 smaller solutions, including many triplets, the smallest being {141, 155, 161}} producing 191
and the biggest is {171270717,190300795,199815833}, producing the prime 228360959.
There are also many quadruples, quintuples, ... and so on.
The biggest set of semiprimes with the same prime f-value is :
{129729595,142702549,144555827,149189017,150810629,152432233,153729497,153945703,154233973,154260179,154563413,154819979,154864519,154980317,155010007,155057509,155321273,155412847,155438989,155482519,155489353,155493223,155494477,155514389,155531869,155547929,155550709,155565227,155565973,155589433,155594917,155599699,155603783,155604613,155605819,155621017,155621897,155623213,155630557,155640757,155645977,155646553,155647243,155649959,155649997,155650207,155650219,155650549,155650567} (49 members).
All these semiprimes have f-value 155675519.

I could not resist the temptation to look for consecutive semiprimes (with the same parity) with the same prime f-value.
Here are those I found :
{33,35}, f-value : 47
{25019,25021}}, f-value : 25343
{205727,205729}}, f-value : 206639
{1887239,1887241}}, f-value : 1889999
{4632491,4632493}}, f-value : 4636799
{28475387,28475389}}, f-value : 28486079
{173112767,173112769}}, f-value : 173139119
{184865201,184865203}}, f-value : 184892399.

***

Jean-Marc wrote:

Q1. Larger pair giving the same prime result :

[332, 377] -> 419
351 = (3^3 * 13) and 351 + 3 * 3 + 13 = 373
352 = (2^5 * 11) and 352 + 2 * 5 + 11 = 373

[351, 352] -> 373
351 = (3^3 * 13) and 351 + 3 * 3 + 13 = 373
352 = (2^5 * 11) and 352 + 2 * 5 + 11 = 373

[14, 15] -> 23
14 = (2 * 7) and 14 + 2 + 7 = 23
15 = (3 * 5) and 15 + 3 + 5 = 23

[116, 135] -> 149
116 = (2^2 * 29) and 116 + 2 * 2 + 29 = 149
135 = (3^3 * 5) and 135 + 3 * 3 + 5 = 149

[1028, 1254] -> 1289
1028 = (2^2 * 257) and 1028 + 2 * 2 + 257 = 1289
1254 = (2 * 3 * 11 * 19) and 1254 + 2 + 3 + 11 + 19 = 1289

[10057, 10146] -> 10259
10057 = (89 * 113) and 10057 + 89 + 113 = 10259
10146 = (2 * 3 * 19 * 89) and 10146 + 2 + 3 + 19 + 89 = 10259

[100086, 101166] -> 102481
100086 = (2 * 3 * 7 * 2383) and 100086 + 2 + 3 + 7 + 2383 = 102481
101166 = (2 * 3 * 13 * 1297) and 101166 + 2 + 3 + 13 + 1297 = 102481

[1000148, 1249797] -> 1250189
1000148 = (2^2 * 250037) and 1000148 + 2 * 2 + 250037 = 1250189
1249797 = (3 * 23 * 59 * 307) and 1249797 + 3 + 23 + 59 + 307 = 1250189

[10000020, 10098773] -> 10166699
10000020 = (2^2 * 3 * 5 * 166667) and 10000020 + 2 * 2 + 3 + 5 + 166667 = 10166699
10098773 = (149 * 67777) and 10098773 + 149 + 67777 = 10166699

[100000250, 100055333] -> 100057417
100000250 = (2 * 5^3 * 7 * 57143) and 100000250 + 2 + 5 * 3 + 7 + 57143 = 100057417
100055333 = (7 * 73 * 103 * 1901) and 100055333 + 7 + 73 + 103 + 1901 = 100057417

[1000000047, 1328224820] -> 1333333399
1000000047 = (3 * 333333349) and 1000000047 + 3 + 333333349 = 1333333399
1328224820 = (2^2 * 5 * 13 * 5108557) and 1328224820 + 2 * 2 + 5 + 13 + 5108557 = 1333333399

[10000000031, 11428566646] -> 11428571471
10000000031 = (7 * 1428571433) and 10000000031 + 7 + 1428571433 = 11428571471
11428566646 = (2 * 7^2 * 23 * 1583 * 3203) and 11428566646 + 2 + 7 * 2 + 23 + 1583 + 3203 = 11428571471

Q2. More than two to give the same prime result :

n solutions -> p

3 [38, 44, 48] -> 59
38 = (2 * 19) and 38 + 2 + 19 = 59
44 = (2^2 * 11) and 44 + 2 * 2 + 11 = 59
48 = (2^4 * 3) and 48 + 2 * 4 + 3 = 59

5 [158, 177, 188, 203, 209] -> 239
158 = (2 * 79) and 158 + 2 + 79 = 239
177 = (3 * 59) and 177 + 3 + 59 = 239
188 = (2^2 * 47) and 188 + 2 * 2 + 47 = 239
203 = (7 * 29) and 203 + 7 + 29 = 239
209 = (11 * 19) and 209 + 11 + 19 = 239

8 [958, 1077, 1195, 1253, 1343, 1349, 1357, 1363] -> 1439
958 = (2 * 479) and 958 + 2 + 479 = 1439
1077 = (3 * 359) and 1077 + 3 + 359 = 1439
1195 = (5 * 239) and 1195 + 5 + 239 = 1439
1253 = (7 * 179) and 1253 + 7 + 179 = 1439
1343 = (17 * 79) and 1343 + 17 + 79 = 1439
1349 = (19 * 71) and 1349 + 19 + 71 = 1439
1357 = (23 * 59) and 1357 + 23 + 59 = 1439
1363 = (29 * 47) and 1363 + 29 + 47 = 1439

9 [5667, 6044, 6295, 7123, 7279, 7339, 7367, 7387, 7482] -> 7559
5667 = (3 * 1889) and 5667 + 3 + 1889 = 7559
6044 = (2^2 * 1511) and 6044 + 2 * 2 + 1511 = 7559
6295 = (5 * 1259) and 6295 + 5 + 1259 = 7559
7123 = (17 * 419) and 7123 + 17 + 419 = 7559
7279 = (29 * 251) and 7279 + 29 + 251 = 7559
7339 = (41 * 179) and 7339 + 41 + 179 = 7559
7367 = (53 * 139) and 7367 + 53 + 139 = 7559
7387 = (83 * 89) and 7387 + 83 + 89 = 7559
7482 = (2 * 3 * 29 * 43) and 7482 + 2 + 3 + 29 + 43 = 7559

10 [15118, 17007, 18895, 20779, 21047, 21403, 22207, 22327, 22339, 22458] -> 22679
15118 = (2 * 7559) and 15118 + 2 + 7559 = 22679
17007 = (3 * 5669) and 17007 + 3 + 5669 = 22679
18895 = (5 * 3779) and 18895 + 5 + 3779 = 22679
20779 = (11 * 1889) and 20779 + 11 + 1889 = 22679
21047 = (13 * 1619) and 21047 + 13 + 1619 = 22679
21403 = (17 * 1259) and 21403 + 17 + 1259 = 22679
22207 = (53 * 419) and 22207 + 53 + 419 = 22679
22327 = (83 * 269) and 22327 + 83 + 269 = 22679
22339 = (89 * 251) and 22339 + 89 + 251 = 22679
22458 = (2 * 3 * 19 * 197) and 22458 + 2 + 3 + 19 + 197 = 22679

11 [30236, 31495, 35087, 35683, 35891, 36511, 37267, 37291, 37343, 37391, 37399] -> 37799
30236 = (2^2 * 7559) and 30236 + 2 * 2 + 7559 = 37799
31495 = (5 * 6299) and 31495 + 5 + 6299 = 37799
35087 = (13 * 2699) and 35087 + 13 + 2699 = 37799
35683 = (17 * 2099) and 35683 + 17 + 2099 = 37799
35891 = (19 * 1889) and 35891 + 19 + 1889 = 37799
36511 = (29 * 1259) and 36511 + 29 + 1259 = 37799
37267 = (83 * 449) and 37267 + 83 + 449 = 37799
37291 = (89 * 419) and 37291 + 89 + 419 = 37799
37343 = (107 * 349) and 37343 + 107 + 349 = 37799
37391 = (139 * 269) and 37391 + 139 + 269 = 37799
37399 = (149 * 251) and 37399 + 149 + 251 = 37799

14 [41577, 44348, 46195, 52343, 53107, 53563, 54079, 54137, 54138, 54599, 54697, 54827, 54889, 54953] -> 55439
41577 = (3 * 13859) and 41577 + 3 + 13859 = 55439
44348 = (2^2 * 11087) and 44348 + 2 * 2 + 11087 = 55439
46195 = (5 * 9239) and 46195 + 5 + 9239 = 55439
52343 = (17 * 3079) and 52343 + 17 + 3079 = 55439
53107 = (23 * 2309) and 53107 + 23 + 2309 = 55439
53563 = (29 * 1847) and 53563 + 29 + 1847 = 55439
54079 = (41 * 1319) and 54079 + 41 + 1319 = 55439
54137 = (43 * 1259) and 54137 + 43 + 1259 = 55439
54138 = (2 * 3 * 7 * 1289) and 54138 + 2 + 3 + 7 + 1289 = 55439
54599 = (71 * 769) and 54599 + 71 + 769 = 55439
54697 = (83 * 659) and 54697 + 83 + 659 = 55439
54827 = (109 * 503) and 54827 + 109 + 503 = 55439
54889 = (131 * 419) and 54889 + 131 + 419 = 55439
54953 = (179 * 307) and 54953 + 179 + 307 = 55439

17 [110878, 124737, 145523, 152449, 157063, 162497, 163187, 163939, 164257, 164383, 164699, 164929, 164993, 165283, 165409, 165499, 166257] -> 166319
110878 = (2 * 55439) and 110878 + 2 + 55439 = 166319
124737 = (3 * 41579) and 124737 + 3 + 41579 = 166319
145523 = (7 * 20789) and 145523 + 7 + 20789 = 166319
152449 = (11 * 13859) and 152449 + 11 + 13859 = 166319
157063 = (17 * 9239) and 157063 + 17 + 9239 = 166319
162497 = (43 * 3779) and 162497 + 43 + 3779 = 166319
163187 = (53 * 3079) and 163187 + 53 + 3079 = 166319
163939 = (71 * 2309) and 163939 + 71 + 2309 = 166319
164257 = (83 * 1979) and 164257 + 83 + 1979 = 166319
164383 = (89 * 1847) and 164383 + 89 + 1847 = 166319
164699 = (109 * 1511) and 164699 + 109 + 1511 = 166319
164929 = (131 * 1259) and 164929 + 131 + 1259 = 166319
164993 = (139 * 1187) and 164993 + 139 + 1187 = 166319
165283 = (197 * 839) and 165283 + 197 + 839 = 166319
165409 = (251 * 659) and 165409 + 251 + 659 = 166319
165499 = (359 * 461) and 165499 + 359 + 461 = 166319
166257 = (3^2 * 7^2 * 13 * 29) and 166257 + 3 * 2 + 7 * 2 + 13 + 29 = 166319

18 [385557, 471229, 485503, 488357, 492637, 501799, 503323, 506453, 508279, 508939, 509213, 510269, 511699, 511789, 511919, 512293, 512437, 512557] -> 514079
385557 = (3 * 128519) and 385557 + 3 + 128519 = 514079
471229 = (11 * 42839) and 471229 + 11 + 42839 = 514079
485503 = (17 * 28559) and 485503 + 17 + 28559 = 514079
488357 = (19 * 25703) and 488357 + 19 + 25703 = 514079
492637 = (23 * 21419) and 492637 + 23 + 21419 = 514079
501799 = (41 * 12239) and 501799 + 41 + 12239 = 514079
503323 = (47 * 10709) and 503323 + 47 + 10709 = 514079
506453 = (67 * 7559) and 506453 + 67 + 7559 = 514079
508279 = (89 * 5711) and 508279 + 89 + 5711 = 514079
508939 = (101 * 5039) and 508939 + 101 + 5039 = 514079
509213 = (107 * 4759) and 509213 + 107 + 4759 = 514079
510269 = (139 * 3671) and 510269 + 139 + 3671 = 514079
511699 = (239 * 2141) and 511699 + 239 + 2141 = 514079
511789 = (251 * 2039) and 511789 + 251 + 2039 = 514079
511919 = (271 * 1889) and 511919 + 271 + 1889 = 514079
512293 = (359 * 1427) and 512293 + 359 + 1427 = 514079
512437 = (419 * 1223) and 512437 + 419 + 1223 = 514079
512557 = (503 * 1019) and 512557 + 503 + 1019 = 514079

19 [498957, 554395, 609829, 644459, 651373, 654133, 655969, 657277, 659123, 660109, 660389, 661723, 662087, 663073, 663277, 663329, 663379, 663457, 663647] -> 665279
498957 = (3 * 166319) and 498957 + 3 + 166319 = 665279
554395 = (5 * 110879) and 554395 + 5 + 110879 = 665279
609829 = (11 * 55439) and 609829 + 11 + 55439 = 665279
644459 = (31 * 20789) and 644459 + 31 + 20789 = 665279
651373 = (47 * 13859) and 651373 + 47 + 13859 = 665279
654133 = (59 * 11087) and 654133 + 59 + 11087 = 665279
655969 = (71 * 9239) and 655969 + 71 + 9239 = 665279
657277 = (83 * 7919) and 657277 + 83 + 7919 = 665279
659123 = (109 * 6047) and 659123 + 109 + 6047 = 665279
660109 = (131 * 5039) and 660109 + 131 + 5039 = 665279
660389 = (139 * 4751) and 660389 + 139 + 4751 = 665279
661723 = (197 * 3359) and 661723 + 197 + 3359 = 665279
662087 = (223 * 2969) and 662087 + 223 + 2969 = 665279
663073 = (359 * 1847) and 663073 + 359 + 1847 = 665279
663277 = (419 * 1583) and 663277 + 419 + 1583 = 665279
663329 = (439 * 1511) and 663329 + 439 + 1511 = 665279
663379 = (461 * 1439) and 663379 + 461 + 1439 = 665279
663457 = (503 * 1319) and 663457 + 503 + 1319 = 665279
663647 = (769 * 863) and 663647 + 769 + 863 = 665279

22 [604798, 680397, 725756, 869377, 878819, 885559, 888253, 896317, 897031, 901003, 901633, 901981, 902927, 903181, 903571, 904259, 904669, 904831, 905089, 905153, 905221, 905287] -> 907199
604798 = (2 * 302399) and 604798 + 2 + 302399 = 907199
680397 = (3 * 226799) and 680397 + 3 + 226799 = 907199
725756 = (2^2 * 181439) and 725756 + 2 * 2 + 181439 = 907199
869377 = (23 * 37799) and 869377 + 23 + 37799 = 907199
878819 = (31 * 28349) and 878819 + 31 + 28349 = 907199
885559 = (41 * 21599) and 885559 + 41 + 21599 = 907199
888253 = (47 * 18899) and 888253 + 47 + 18899 = 907199
896317 = (83 * 10799) and 896317 + 83 + 10799 = 907199
897031 = (89 * 10079) and 897031 + 89 + 10079 = 907199
901003 = (149 * 6047) and 901003 + 149 + 6047 = 907199
901633 = (167 * 5399) and 901633 + 167 + 5399 = 907199
901981 = (179 * 5039) and 901981 + 179 + 5039 = 907199
902927 = (223 * 4049) and 902927 + 223 + 4049 = 907199
903181 = (239 * 3779) and 903181 + 239 + 3779 = 907199
903571 = (269 * 3359) and 903571 + 269 + 3359 = 907199
904259 = (349 * 2591) and 904259 + 349 + 2591 = 907199
904669 = (431 * 2099) and 904669 + 431 + 2099 = 907199
904831 = (479 * 1889) and 904831 + 479 + 1889 = 907199
905089 = (599 * 1511) and 905089 + 599 + 1511 = 907199
905153 = (647 * 1399) and 905153 + 647 + 1399 = 907199
905221 = (719 * 1259) and 905221 + 719 + 1259 = 907199
905287 = (863 * 1049) and 905287 + 863 + 1049 = 907199

24 [665276, 692995, 772187, 785383, 790001, 796927, 812657, 817681, 819979, 822271, 823793, 823931, 825169, 825521, 827243, 828049, 828251, 828593, 828931, 829201, 829271, 829303, 829607, 829681] -> 831599
665276 = (2^2 * 166319) and 665276 + 2 * 2 + 166319 = 831599
692995 = (5 * 138599) and 692995 + 5 + 138599 = 831599
772187 = (13 * 59399) and 772187 + 13 + 59399 = 831599
785383 = (17 * 46199) and 785383 + 17 + 46199 = 831599
790001 = (19 * 41579) and 790001 + 19 + 41579 = 831599
796927 = (23 * 34649) and 796927 + 23 + 34649 = 831599
812657 = (43 * 18899) and 812657 + 43 + 18899 = 831599
817681 = (59 * 13859) and 817681 + 59 + 13859 = 831599
819979 = (71 * 11549) and 819979 + 71 + 11549 = 831599
822271 = (89 * 9239) and 822271 + 89 + 9239 = 831599
823793 = (107 * 7699) and 823793 + 107 + 7699 = 831599
823931 = (109 * 7559) and 823931 + 109 + 7559 = 831599
825169 = (131 * 6299) and 825169 + 131 + 6299 = 831599
825521 = (139 * 5939) and 825521 + 139 + 5939 = 831599
827243 = (199 * 4157) and 827243 + 199 + 4157 = 831599
828049 = (251 * 3299) and 828049 + 251 + 3299 = 831599
828251 = (269 * 3079) and 828251 + 269 + 3079 = 831599
828593 = (307 * 2699) and 828593 + 307 + 2699 = 831599
828931 = (359 * 2309) and 828931 + 359 + 2309 = 831599
829201 = (419 * 1979) and 829201 + 419 + 1979 = 831599
829271 = (439 * 1889) and 829271 + 439 + 1889 = 831599
829303 = (449 * 1847) and 829303 + 449 + 1847 = 831599
829607 = (593 * 1399) and 829607 + 593 + 1399 = 831599
829681 = (659 * 1259) and 829681 + 659 + 1259 = 831599

29 [3511198, 3950097, 4608443, 4827889, 4890587, 5128163, 5141359, 5147057, 5220487, 5229041, 5231539, 5231999, 5243473, 5245649, 5246587, 5249393, 5251811, 5252561, 5253841, 5254391, 5254939, 5256991, 5258417, 5259403, 5259691, 5261491, 5261639, 5262011, 5262161] -> 5266799
3511198 = (2 * 1755599) and 3511198 + 2 + 1755599 = 5266799
3950097 = (3 * 1316699) and 3950097 + 3 + 1316699 = 5266799
4608443 = (7 * 658349) and 4608443 + 7 + 658349 = 5266799
4827889 = (11 * 438899) and 4827889 + 11 + 438899 = 5266799
4890587 = (13 * 376199) and 4890587 + 13 + 376199 = 5266799
5128163 = (37 * 138599) and 5128163 + 37 + 138599 = 5266799
5141359 = (41 * 125399) and 5141359 + 41 + 125399 = 5266799
5147057 = (43 * 119699) and 5147057 + 43 + 119699 = 5266799
5220487 = (113 * 46199) and 5220487 + 113 + 46199 = 5266799
5229041 = (139 * 37619) and 5229041 + 139 + 37619 = 5266799
5231539 = (149 * 35111) and 5231539 + 149 + 35111 = 5266799
5231999 = (151 * 34649) and 5231999 + 151 + 34649 = 5266799
5243473 = (227 * 23099) and 5243473 + 227 + 23099 = 5266799
5245649 = (251 * 20899) and 5245649 + 251 + 20899 = 5266799
5246587 = (263 * 19949) and 5246587 + 263 + 19949 = 5266799
5249393 = (307 * 17099) and 5249393 + 307 + 17099 = 5266799
5251811 = (359 * 14629) and 5251811 + 359 + 14629 = 5266799
5252561 = (379 * 13859) and 5252561 + 379 + 13859 = 5266799
5253841 = (419 * 12539) and 5253841 + 419 + 12539 = 5266799
5254391 = (439 * 11969) and 5254391 + 439 + 11969 = 5266799
5254939 = (461 * 11399) and 5254939 + 461 + 11399 = 5266799
5256991 = (569 * 9239) and 5256991 + 569 + 9239 = 5266799
5258417 = (683 * 7699) and 5258417 + 683 + 7699 = 5266799
5259403 = (797 * 6599) and 5259403 + 797 + 6599 = 5266799
5259691 = (839 * 6269) and 5259691 + 839 + 6269 = 5266799
5261491 = (1319 * 3989) and 5261491 + 1319 + 3989 = 5266799
5261639 = (1399 * 3761) and 5261639 + 1399 + 3761 = 5266799
5262011 = (1709 * 3079) and 5262011 + 1709 + 3079 = 5266799
5262161 = (1979 * 2659) and 5262161 + 1979 + 2659 = 5266799

30 [4864857, 5405395, 5675663, 5945929, 6023147, 6216187, 6331999, 6339017, 6378313, 6396319, 6427403, 6437209, 6447703, 6462187, 6465113, 6465379, 6469459, 6470617, 6471299, 6472153, 6475823, 6476539, 6477287, 6478063, 6479449, 6480073, 6480679, 6480763, 6480857, 6485325] -> 6486479
4864857 = (3 * 1621619) and 4864857 + 3 + 1621619 = 6486479
5405395 = (5 * 1081079) and 5405395 + 5 + 1081079 = 6486479
5675663 = (7 * 810809) and 5675663 + 7 + 810809 = 6486479
5945929 = (11 * 540539) and 5945929 + 11 + 540539 = 6486479
6023147 = (13 * 463319) and 6023147 + 13 + 463319 = 6486479
6216187 = (23 * 270269) and 6216187 + 23 + 270269 = 6486479
6331999 = (41 * 154439) and 6331999 + 41 + 154439 = 6486479
6339017 = (43 * 147419) and 6339017 + 43 + 147419 = 6486479
6378313 = (59 * 108107) and 6378313 + 59 + 108107 = 6486479
6396319 = (71 * 90089) and 6396319 + 71 + 90089 = 6486479
6427403 = (109 * 58967) and 6427403 + 109 + 58967 = 6486479
6437209 = (131 * 49139) and 6437209 + 131 + 49139 = 6486479
6447703 = (167 * 38609) and 6447703 + 167 + 38609 = 6486479
6462187 = (269 * 24023) and 6462187 + 269 + 24023 = 6486479
6465113 = (307 * 21059) and 6465113 + 307 + 21059 = 6486479
6465379 = (311 * 20789) and 6465379 + 311 + 20789 = 6486479
6469459 = (389 * 16631) and 6469459 + 389 + 16631 = 6486479
6470617 = (419 * 15443) and 6470617 + 419 + 15443 = 6486479
6471299 = (439 * 14741) and 6471299 + 439 + 14741 = 6486479
6472153 = (467 * 13859) and 6472153 + 467 + 13859 = 6486479
6475823 = (647 * 10009) and 6475823 + 647 + 10009 = 6486479
6476539 = (701 * 9239) and 6476539 + 701 + 9239 = 6486479
6477287 = (769 * 8423) and 6477287 + 769 + 8423 = 6486479
6478063 = (857 * 7559) and 6478063 + 857 + 7559 = 6486479
6479449 = (1091 * 5939) and 6479449 + 1091 + 5939 = 6486479
6480073 = (1259 * 5147) and 6480073 + 1259 + 5147 = 6486479
6480679 = (1511 * 4289) and 6480679 + 1511 + 4289 = 6486479
6480763 = (1559 * 4157) and 6480763 + 1559 + 4157 = 6486479
6480857 = (1619 * 4003) and 6480857 + 1619 + 4003 = 6486479
6485325 = (3 * 5^2 * 7 * 11 * 1123) and 6485325 + 3 + 5 * 2 + 7 + 11 + 1123 = 6486479

32 [5543998, 7900181, 8126957, 8142703, 8177341, 8200429, 8216917, 8223511, 8256461, 8260411, 8266333, 8266776, 8269621, 8273803, 8274221, 8281111, 8282749, 8292541, 8296319, 8296661, 8298869, 8301407, 8301541, 8303731, 8304431, 8307031, 8308141, 8308381, 8308661, 8309843, 8310221, 8310227] -> 8315999
5543998 = (2 * 2771999) and 5543998 + 2 + 2771999 = 8315999
7900181 = (19 * 415799) and 7900181 + 19 + 415799 = 8315999
8126957 = (43 * 188999) and 8126957 + 43 + 188999 = 8315999
8142703 = (47 * 173249) and 8142703 + 47 + 173249 = 8315999
8177341 = (59 * 138599) and 8177341 + 59 + 138599 = 8315999
8200429 = (71 * 115499) and 8200429 + 71 + 115499 = 8315999
8216917 = (83 * 98999) and 8216917 + 83 + 98999 = 8315999
8223511 = (89 * 92399) and 8223511 + 89 + 92399 = 8315999
8256461 = (139 * 59399) and 8256461 + 139 + 59399 = 8315999
8260411 = (149 * 55439) and 8260411 + 149 + 55439 = 8315999
8266333 = (167 * 49499) and 8266333 + 167 + 49499 = 8315999
8266776 = (2^3 * 3 * 7 * 49207) and 8266776 + 2 * 3 + 3 + 7 + 49207 = 8315999
8269621 = (179 * 46199) and 8269621 + 179 + 46199 = 8315999
8273803 = (197 * 41999) and 8273803 + 197 + 41999 = 8315999
8274221 = (199 * 41579) and 8274221 + 199 + 41579 = 8315999
8281111 = (239 * 34649) and 8281111 + 239 + 34649 = 8315999
8282749 = (251 * 32999) and 8282749 + 251 + 32999 = 8315999
8292541 = (359 * 23099) and 8292541 + 359 + 23099 = 8315999
8296319 = (431 * 19249) and 8296319 + 431 + 19249 = 8315999
8296661 = (439 * 18899) and 8296661 + 439 + 18899 = 8315999
8298869 = (499 * 16631) and 8298869 + 499 + 16631 = 8315999
8301407 = (593 * 13999) and 8301407 + 593 + 13999 = 8315999
8301541 = (599 * 13859) and 8301541 + 599 + 13859 = 8315999
8303731 = (719 * 11549) and 8303731 + 719 + 11549 = 8315999
8304431 = (769 * 10799) and 8304431 + 769 + 10799 = 8315999
8307031 = (1049 * 7919) and 8307031 + 1049 + 7919 = 8315999
8308141 = (1259 * 6599) and 8308141 + 1259 + 6599 = 8315999
8308381 = (1319 * 6299) and 8308381 + 1319 + 6299 = 8315999
8308661 = (1399 * 5939) and 8308661 + 1399 + 5939 = 8315999
8309843 = (1999 * 4157) and 8309843 + 1999 + 4157 = 8315999
8310221 = (2699 * 3079) and 8310221 + 2699 + 3079 = 8315999
8310227 = (2749 * 3023) and 8310227 + 2749 + 3023 = 8315999

34 [13097697, 13970876, 14552995, 15280643, 16881451, 17047759, 17304731, 17331169, 17338721, 17347027, 17376083, 17398651, 17403907, 17421601, 17425339, 17428447, 17433607, 17436481, 17440151, 17441231, 17441971, 17445919, 17447713, 17448481, 17449051, 17449727, 17450539, 17452471, 17452801, 17453633, 17453731, 17454721, 17454751, 17454851]
13097697 = (3 * 4365899) and 13097697 + 3 + 4365899 = 17463599
13970876 = (2^2 * 3492719) and 13970876 + 2 * 2 + 3492719 = 17463599
14552995 = (5 * 2910599) and 14552995 + 5 + 2910599 = 17463599
15280643 = (7 * 2182949) and 15280643 + 7 + 2182949 = 17463599
16881451 = (29 * 582119) and 16881451 + 29 + 582119 = 17463599
17047759 = (41 * 415799) and 17047759 + 41 + 415799 = 17463599
17304731 = (109 * 158759) and 17304731 + 109 + 158759 = 17463599
17331169 = (131 * 132299) and 17331169 + 131 + 132299 = 17463599
17338721 = (139 * 124739) and 17338721 + 139 + 124739 = 17463599
17347027 = (149 * 116423) and 17347027 + 149 + 116423 = 17463599
17376083 = (199 * 87317) and 17376083 + 199 + 87317 = 17463599
17398651 = (269 * 64679) and 17398651 + 269 + 64679 = 17463599
17403907 = (293 * 59399) and 17403907 + 293 + 59399 = 17463599
17421601 = (419 * 41579) and 17421601 + 419 + 41579 = 17463599
17425339 = (461 * 37799) and 17425339 + 461 + 37799 = 17463599
17428447 = (503 * 34649) and 17428447 + 503 + 34649 = 17463599
17433607 = (593 * 29399) and 17433607 + 593 + 29399 = 17463599
17436481 = (659 * 26459) and 17436481 + 659 + 26459 = 17463599
17440151 = (769 * 22679) and 17440151 + 769 + 22679 = 17463599
17441231 = (809 * 21559) and 17441231 + 809 + 21559 = 17463599
17441971 = (839 * 20789) and 17441971 + 839 + 20789 = 17463599
17445919 = (1049 * 16631) and 17445919 + 1049 + 16631 = 17463599
17447713 = (1187 * 14699) and 17447713 + 1187 + 14699 = 17463599
17448481 = (1259 * 13859) and 17448481 + 1259 + 13859 = 17463599
17449051 = (1319 * 13229) and 17449051 + 1319 + 13229 = 17463599
17449727 = (1399 * 12473) and 17449727 + 1399 + 12473 = 17463599
17450539 = (1511 * 11549) and 17450539 + 1511 + 11549 = 17463599
17452471 = (1889 * 9239) and 17452471 + 1889 + 9239 = 17463599
17452801 = (1979 * 8819) and 17452801 + 1979 + 8819 = 17463599
17453633 = (2267 * 7699) and 17453633 + 2267 + 7699 = 17463599
17453731 = (2309 * 7559) and 17453731 + 2309 + 7559 = 17463599
17454721 = (2939 * 5939) and 17454721 + 2939 + 5939 = 17463599
17454751 = (2969 * 5879) and 17454751 + 2969 + 5879 = 17463599
17454851 = (3079 * 5669) and 17454851 + 3079 + 5669 = 17463599

43 [21067198, 23700597, 25280636, 26333995, 27650693, 28967389, 29845183, 30284077, 30769163, 30942403, 31015547, 31074061, 31161829, 31224517, 31313411, 31389979, 31442597, 31461973, 31475149, 31480837, 31525141, 31530127, 31531939, 31544791, 31547533, 31553917, 31556191, 31558991, 31562341, 31565239, 31576333, 31576829, 31578389, 31579267, 31581853, 31582861, 31584811, 31587191, 31587461, 31588891, 31589413, 31589491, 31589549] -> 31600799
21067198 = (2 * 10533599) and 21067198 + 2 + 10533599 = 31600799
23700597 = (3 * 7900199) and 23700597 + 3 + 7900199 = 31600799
25280636 = (2^2 * 6320159) and 25280636 + 2 * 2 + 6320159 = 31600799
26333995 = (5 * 5266799) and 26333995 + 5 + 5266799 = 31600799
27650693 = (7 * 3950099) and 27650693 + 7 + 3950099 = 31600799
28967389 = (11 * 2633399) and 28967389 + 11 + 2633399 = 31600799
29845183 = (17 * 1755599) and 29845183 + 17 + 1755599 = 31600799
30284077 = (23 * 1316699) and 30284077 + 23 + 1316699 = 31600799
30769163 = (37 * 831599) and 30769163 + 37 + 831599 = 31600799
30942403 = (47 * 658349) and 30942403 + 47 + 658349 = 31600799
31015547 = (53 * 585199) and 31015547 + 53 + 585199 = 31600799
31074061 = (59 * 526679) and 31074061 + 59 + 526679 = 31600799
31161829 = (71 * 438899) and 31161829 + 71 + 438899 = 31600799
31224517 = (83 * 376199) and 31224517 + 83 + 376199 = 31600799
31313411 = (109 * 287279) and 31313411 + 109 + 287279 = 31600799
31389979 = (149 * 210671) and 31389979 + 149 + 210671 = 31600799
31442597 = (199 * 158003) and 31442597 + 199 + 158003 = 31600799
31461973 = (227 * 138599) and 31461973 + 227 + 138599 = 31600799
31475149 = (251 * 125399) and 31475149 + 251 + 125399 = 31600799
31480837 = (263 * 119699) and 31480837 + 263 + 119699 = 31600799
31525141 = (419 * 75239) and 31525141 + 419 + 75239 = 31600799
31530127 = (449 * 70223) and 31530127 + 449 + 70223 = 31600799
31531939 = (461 * 68399) and 31531939 + 461 + 68399 = 31600799
31544791 = (569 * 55439) and 31544791 + 569 + 55439 = 31600799
31547533 = (599 * 52667) and 31547533 + 599 + 52667 = 31600799
31553917 = (683 * 46199) and 31553917 + 683 + 46199 = 31600799
31556191 = (719 * 43889) and 31556191 + 719 + 43889 = 31600799
31558991 = (769 * 41039) and 31558991 + 769 + 41039 = 31600799
31562341 = (839 * 37619) and 31562341 + 839 + 37619 = 31600799
31565239 = (911 * 34649) and 31565239 + 911 + 34649 = 31600799
31576333 = (1367 * 23099) and 31576333 + 1367 + 23099 = 31600799
31576829 = (1399 * 22571) and 31576829 + 1399 + 22571 = 31600799
31578389 = (1511 * 20899) and 31578389 + 1511 + 20899 = 31600799
31579267 = (1583 * 19949) and 31579267 + 1583 + 19949 = 31600799
31581853 = (1847 * 17099) and 31581853 + 1847 + 17099 = 31600799
31582861 = (1979 * 15959) and 31582861 + 1979 + 15959 = 31600799
31584811 = (2309 * 13679) and 31584811 + 2309 + 13679 = 31600799
31587191 = (2969 * 10639) and 31587191 + 2969 + 10639 = 31600799
31587461 = (3079 * 10259) and 31587461 + 3079 + 10259 = 31600799
31588891 = (3989 * 7919) and 31588891 + 3989 + 7919 = 31600799
31589413 = (4787 * 6599) and 31589413 + 4787 + 6599 = 31600799
31589491 = (5039 * 6269) and 31589491 + 5039 + 6269 = 31600799
31589549 = (5399 * 5851) and 31589549 + 5399 + 5851 = 31600799

46 [51891836, 60231587, 61621541, 63390557, 63783661, 64053911, 64240997, 64401341, 64508219, 64537003, 64594291, 64607149, 64653893, 64679123, 64684261, 64698091, 64709941, 64716941, 64725733, 64755007, 64756093, 64771699, 64773991, 64779791, 64804309, 64812061, 64814341, 64820389, 64821661, 64822267, 64823141, 64827853, 64828279, 64828591, 64834163, 64838077, 64839991, 64840661, 64840891, 64843861, 64846261, 64846399, 64847389, 64847863, 64848541, 64848677] -> 64864799

51891836 = (2^2 * 12972959) and 51891836 + 2 * 2 + 12972959 = 64864799
60231587 = (13 * 4633199) and 60231587 + 13 + 4633199 = 64864799
61621541 = (19 * 3243239) and 61621541 + 19 + 3243239 = 64864799
63390557 = (43 * 1474199) and 63390557 + 43 + 1474199 = 64864799
63783661 = (59 * 1081079) and 63783661 + 59 + 1081079 = 64864799
64053911 = (79 * 810809) and 64053911 + 79 + 810809 = 64864799
64240997 = (103 * 623699) and 64240997 + 103 + 623699 = 64864799
64401341 = (139 * 463319) and 64401341 + 139 + 463319 = 64864799
64508219 = (181 * 356399) and 64508219 + 181 + 356399 = 64864799
64537003 = (197 * 327599) and 64537003 + 197 + 327599 = 64864799
64594291 = (239 * 270269) and 64594291 + 239 + 270269 = 64864799
64607149 = (251 * 257399) and 64607149 + 251 + 257399 = 64864799
64653893 = (307 * 210599) and 64653893 + 307 + 210599 = 64864799
64679123 = (349 * 185327) and 64679123 + 349 + 185327 = 64864799
64684261 = (359 * 180179) and 64684261 + 359 + 180179 = 64864799
64698091 = (389 * 166319) and 64698091 + 389 + 166319 = 64864799
64709941 = (419 * 154439) and 64709941 + 419 + 154439 = 64864799
64716941 = (439 * 147419) and 64716941 + 439 + 147419 = 64864799
64725733 = (467 * 138599) and 64725733 + 467 + 138599 = 64864799
64755007 = (593 * 109199) and 64755007 + 593 + 109199 = 64864799
64756093 = (599 * 108107) and 64756093 + 599 + 108107 = 64864799
64771699 = (701 * 92399) and 64771699 + 701 + 92399 = 64864799
64773991 = (719 * 90089) and 64773991 + 719 + 90089 = 64864799
64779791 = (769 * 84239) and 64779791 + 769 + 84239 = 64864799
64804309 = (1091 * 59399) and 64804309 + 1091 + 59399 = 64864799
64812061 = (1259 * 51479) and 64812061 + 1259 + 51479 = 64864799
64814341 = (1319 * 49139) and 64814341 + 1319 + 49139 = 64864799
64820389 = (1511 * 42899) and 64820389 + 1511 + 42899 = 64864799
64821661 = (1559 * 41579) and 64821661 + 1559 + 41579 = 64864799
64822267 = (1583 * 40949) and 64822267 + 1583 + 40949 = 64864799
64823141 = (1619 * 40039) and 64823141 + 1619 + 40039 = 64864799
64827853 = (1847 * 35099) and 64827853 + 1847 + 35099 = 64864799
64828279 = (1871 * 34649) and 64828279 + 1871 + 34649 = 64864799
64828591 = (1889 * 34319) and 64828591 + 1889 + 34319 = 64864799
64834163 = (2287 * 28349) and 64834163 + 2287 + 28349 = 64864799
64838077 = (2699 * 24023) and 64838077 + 2699 + 24023 = 64864799
64839991 = (2969 * 21839) and 64839991 + 2969 + 21839 = 64864799
64840661 = (3079 * 21059) and 64840661 + 3079 + 21059 = 64864799
64840891 = (3119 * 20789) and 64840891 + 3119 + 20789 = 64864799
64843861 = (3779 * 17159) and 64843861 + 3779 + 17159 = 64864799
64846261 = (4679 * 13859) and 64846261 + 4679 + 13859 = 64864799
64846399 = (4751 * 13649) and 64846399 + 4751 + 13649 = 64864799
64847389 = (5399 * 12011) and 64847389 + 5399 + 12011 = 64864799
64847863 = (5849 * 11087) and 64847863 + 5849 + 11087 = 64864799
64848541 = (7019 * 9239) and 64848541 + 7019 + 9239 = 64864799
64848677 = (7699 * 8423) and 64848677 + 7699 + 8423 = 64864799

49 [129729595, 142702549, 144555827, 149189017, 150810629, 152432233, 153729497, 153945703, 154233973, 154260179, 154563413, 154819979, 154864519, 154980317, 155010007, 155057509, 155321273, 155412847, 155438989, 155482519, 155489353, 155493223, 155494477, 155514389, 155531869, 155547929, 155550709, 155565227, 155565973, 155589433, 155594917, 155599699, 155603783, 155604613, 155605819, 155621017, 155621897, 155623213, 155630557, 155640757, 155645977, 155646553, 155647243, 155649959, 155649997, 155650207, 155650219, 155650549, 155650567] -> 155675519
129729595 = (5 * 25945919) and 129729595 + 5 + 25945919 = 155675519
142702549 = (11 * 12972959) and 142702549 + 11 + 12972959 = 155675519
144555827 = (13 * 11119679) and 144555827 + 13 + 11119679 = 155675519
149189017 = (23 * 6486479) and 149189017 + 23 + 6486479 = 155675519
150810629 = (31 * 4864859) and 150810629 + 31 + 4864859 = 155675519
152432233 = (47 * 3243239) and 152432233 + 47 + 3243239 = 155675519
153729497 = (79 * 1945943) and 153729497 + 79 + 1945943 = 155675519
153945703 = (89 * 1729727) and 153945703 + 89 + 1729727 = 155675519
154233973 = (107 * 1441439) and 154233973 + 107 + 1441439 = 155675519
154260179 = (109 * 1415231) and 154260179 + 109 + 1415231 = 155675519
154563413 = (139 * 1111967) and 154563413 + 139 + 1111967 = 155675519
154819979 = (181 * 855359) and 154819979 + 181 + 855359 = 155675519
154864519 = (191 * 810809) and 154864519 + 191 + 810809 = 155675519
154980317 = (223 * 694979) and 154980317 + 223 + 694979 = 155675519
155010007 = (233 * 665279) and 155010007 + 233 + 665279 = 155675519
155057509 = (251 * 617759) and 155057509 + 251 + 617759 = 155675519
155321273 = (439 * 353807) and 155321273 + 439 + 353807 = 155675519
155412847 = (593 * 262079) and 155412847 + 593 + 262079 = 155675519
155438989 = (659 * 235871) and 155438989 + 659 + 235871 = 155675519
155482519 = (809 * 192191) and 155482519 + 809 + 192191 = 155675519
155489353 = (839 * 185327) and 155489353 + 839 + 185327 = 155675519
155493223 = (857 * 181439) and 155493223 + 857 + 181439 = 155675519
155494477 = (863 * 180179) and 155494477 + 863 + 180179 = 155675519
155514389 = (971 * 160159) and 155514389 + 971 + 160159 = 155675519
155531869 = (1091 * 142559) and 155531869 + 1091 + 142559 = 155675519
155547929 = (1231 * 126359) and 155547929 + 1231 + 126359 = 155675519
155550709 = (1259 * 123551) and 155550709 + 1259 + 123551 = 155675519
155565227 = (1429 * 108863) and 155565227 + 1429 + 108863 = 155675519
155565973 = (1439 * 108107) and 155565973 + 1439 + 108107 = 155675519
155589433 = (1847 * 84239) and 155589433 + 1847 + 84239 = 155675519
155594917 = (1979 * 78623) and 155594917 + 1979 + 78623 = 155675519
155599699 = (2111 * 73709) and 155599699 + 2111 + 73709 = 155675519
155603783 = (2239 * 69497) and 155603783 + 2239 + 69497 = 155675519
155604613 = (2267 * 68639) and 155604613 + 2267 + 68639 = 155675519
155605819 = (2309 * 67391) and 155605819 + 2309 + 67391 = 155675519
155621017 = (3023 * 51479) and 155621017 + 3023 + 51479 = 155675519
155621897 = (3079 * 50543) and 155621897 + 3079 + 50543 = 155675519
155623213 = (3167 * 49139) and 155623213 + 3167 + 49139 = 155675519
155630557 = (3779 * 41183) and 155630557 + 3779 + 41183 = 155675519
155640757 = (5279 * 29483) and 155640757 + 5279 + 29483 = 155675519
155645977 = (6863 * 22679) and 155645977 + 6863 + 22679 = 155675519
155646553 = (7127 * 21839) and 155646553 + 7127 + 21839 = 155675519
155647243 = (7487 * 20789) and 155647243 + 7487 + 20789 = 155675519
155649959 = (10009 * 15551) and 155649959 + 10009 + 15551 = 155675519
155649997 = (10079 * 15443) and 155649997 + 10079 + 15443 = 155675519
155650207 = (10529 * 14783) and 155650207 + 10529 + 14783 = 155675519
155650219 = (10559 * 14741) and 155650219 + 10559 + 14741 = 155675519
155650549 = (12011 * 12959) and 155650549 + 12011 + 12959 = 155675519
155650567 = (12473 * 12479) and 155650567 + 12473 + 12479 = 155675519

51 [205405197, 228227995, 251050789, 254311187, 265315019, 268167853, 269308981, 270450101, 271240097, 271383731, 271917221, 272243233, 272490203, 272732221, 273112481, 273170971, 273221101, 273287933, 273302551, 273457981, 273496673, 273517151, 273529603, 273572389, 273615137, 273621709, 273664801, 273680651, 273711731, 273716231, 273723553, 273725429, 273733301, 273751613, 273757087, 273778237, 273781601, 273788357, 273788731, 273800971, 273801197, 273805471, 273816451, 273821473, 273838177, 273839221, 273839597, 273839887, 273839981, 273840251, 273840481] -> 273873599

205405197 = (3 * 68468399) and 205405197 + 3 + 68468399 = 273873599
228227995 = (5 * 45645599) and 228227995 + 5 + 45645599 = 273873599
251050789 = (11 * 22822799) and 251050789 + 11 + 22822799 = 273873599
254311187 = (13 * 19562399) and 254311187 + 13 + 19562399 = 273873599
265315019 = (31 * 8558549) and 265315019 + 31 + 8558549 = 273873599
268167853 = (47 * 5705699) and 268167853 + 47 + 5705699 = 273873599
269308981 = (59 * 4564559) and 269308981 + 59 + 4564559 = 273873599
270450101 = (79 * 3423419) and 270450101 + 79 + 3423419 = 273873599
271240097 = (103 * 2633399) and 271240097 + 103 + 2633399 = 273873599
271383731 = (109 * 2489759) and 271383731 + 109 + 2489759 = 273873599
271917221 = (139 * 1956239) and 271917221 + 139 + 1956239 = 273873599
272243233 = (167 * 1630199) and 272243233 + 167 + 1630199 = 273873599
272490203 = (197 * 1383199) and 272490203 + 197 + 1383199 = 273873599
272732221 = (239 * 1141139) and 272732221 + 239 + 1141139 = 273873599
273112481 = (359 * 760759) and 273112481 + 359 + 760759 = 273873599
273170971 = (389 * 702239) and 273170971 + 389 + 702239 = 273873599
273221101 = (419 * 652079) and 273221101 + 419 + 652079 = 273873599
273287933 = (467 * 585199) and 273287933 + 467 + 585199 = 273873599
273302551 = (479 * 570569) and 273302551 + 479 + 570569 = 273873599
273457981 = (659 * 414959) and 273457981 + 659 + 414959 = 273873599
273496673 = (727 * 376199) and 273496673 + 727 + 376199 = 273873599
273517151 = (769 * 355679) and 273517151 + 769 + 355679 = 273873599
273529603 = (797 * 343199) and 273529603 + 797 + 343199 = 273873599
273572389 = (911 * 300299) and 273572389 + 911 + 300299 = 273873599
273615137 = (1063 * 257399) and 273615137 + 1063 + 257399 = 273873599
273621709 = (1091 * 250799) and 273621709 + 1091 + 250799 = 273873599
273664801 = (1319 * 207479) and 273664801 + 1319 + 207479 = 273873599
273680651 = (1429 * 191519) and 273680651 + 1429 + 191519 = 273873599
273711731 = (1709 * 160159) and 273711731 + 1709 + 160159 = 273873599
273716231 = (1759 * 155609) and 273716231 + 1759 + 155609 = 273873599
273723553 = (1847 * 148199) and 273723553 + 1847 + 148199 = 273873599
273725429 = (1871 * 146299) and 273725429 + 1871 + 146299 = 273873599
273733301 = (1979 * 138319) and 273733301 + 1979 + 138319 = 273873599
273751613 = (2287 * 119699) and 273751613 + 2287 + 119699 = 273873599
273757087 = (2399 * 114113) and 273757087 + 2399 + 114113 = 273873599
273778237 = (2963 * 92399) and 273778237 + 2963 + 92399 = 273873599
273781601 = (3079 * 88919) and 273781601 + 3079 + 88919 = 273873599
273788357 = (3343 * 81899) and 273788357 + 3343 + 81899 = 273873599
273788731 = (3359 * 81509) and 273788731 + 3359 + 81509 = 273873599
273800971 = (3989 * 68639) and 273800971 + 3989 + 68639 = 273873599
273801197 = (4003 * 68399) and 273801197 + 4003 + 68399 = 273873599
273805471 = (4289 * 63839) and 273805471 + 4289 + 63839 = 273873599
273816451 = (5279 * 51869) and 273816451 + 5279 + 51869 = 273873599
273821473 = (5927 * 46199) and 273821473 + 5927 + 46199 = 273873599
273838177 = (11399 * 24023) and 273838177 + 11399 + 24023 = 273873599
273839221 = (12539 * 21839) and 273839221 + 12539 + 21839 = 273873599
273839597 = (13103 * 20899) and 273839597 + 13103 + 20899 = 273873599
273839887 = (13649 * 20063) and 273839887 + 13649 + 20063 = 273873599
273839981 = (13859 * 19759) and 273839981 + 13859 + 19759 = 273873599
273840251 = (14629 * 18719) and 273840251 + 14629 + 18719 = 273873599
273840481 = (15959 * 17159) and 273840481 + 15959 + 17159 = 273873599

58 [273873598, 328648316, 342341995, 381466787, 387987583, 390269861, 402251803, 403963501, 405104629, 405675191, 406245751, 406860197, 409054567, 409179949, 409476293, 409493389, 409636307, 409668901, 409728941, 409897039, 410125117, 410224499, 410239111, 410245373, 410418103, 410433109, 410475719, 410508733, 410521691, 410557963, 410597779, 410600941, 410628563, 410630251, 410653301, 410668837, 410669111, 410673941, 410684207, 410688143, 410697439, 410723851, 410734327, 410734349, 410738611, 410750611, 410759351, 410760101, 410761151, 410761733, 410762263, 410763589, 410764111, 410766691, 410769277, 410769517, 410769629, 410769851] -> 410810399
273873598 = (2 * 136936799) and 273873598 + 2 + 136936799 = 410810399
328648316 = (2^2 * 82162079) and 328648316 + 2 * 2 + 82162079 = 410810399
342341995 = (5 * 68468399) and 342341995 + 5 + 68468399 = 410810399
381466787 = (13 * 29343599) and 381466787 + 13 + 29343599 = 410810399
387987583 = (17 * 22822799) and 387987583 + 17 + 22822799 = 410810399
390269861 = (19 * 20540519) and 390269861 + 19 + 20540519 = 410810399
402251803 = (47 * 8558549) and 402251803 + 47 + 8558549 = 410810399
403963501 = (59 * 6846839) and 403963501 + 59 + 6846839 = 410810399
405104629 = (71 * 5705699) and 405104629 + 71 + 5705699 = 410810399
405675191 = (79 * 5135129) and 405675191 + 79 + 5135129 = 410810399
406245751 = (89 * 4564559) and 406245751 + 89 + 4564559 = 410810399
406860197 = (103 * 3950099) and 406860197 + 103 + 3950099 = 410810399
409054567 = (233 * 1755599) and 409054567 + 233 + 1755599 = 410810399
409179949 = (251 * 1630199) and 409179949 + 251 + 1630199 = 410810399
409476293 = (307 * 1333799) and 409476293 + 307 + 1333799 = 410810399
409493389 = (311 * 1316699) and 409493389 + 311 + 1316699 = 410810399
409636307 = (349 * 1173743) and 409636307 + 349 + 1173743 = 410810399
409668901 = (359 * 1141139) and 409668901 + 359 + 1141139 = 410810399
409728941 = (379 * 1081079) and 409728941 + 379 + 1081079 = 410810399
409897039 = (449 * 912911) and 409897039 + 449 + 912911 = 410810399
410125117 = (599 * 684683) and 410125117 + 599 + 684683 = 410810399
410224499 = (701 * 585199) and 410224499 + 701 + 585199 = 410810399
410239111 = (719 * 570569) and 410239111 + 719 + 570569 = 410810399
410245373 = (727 * 564299) and 410245373 + 727 + 564299 = 410810399
410418103 = (1049 * 391247) and 410418103 + 1049 + 391247 = 410810399
410433109 = (1091 * 376199) and 410433109 + 1091 + 376199 = 410810399
410475719 = (1231 * 333449) and 410475719 + 1231 + 333449 = 410810399
410508733 = (1367 * 300299) and 410508733 + 1367 + 300299 = 410810399
410521691 = (1429 * 287279) and 410521691 + 1429 + 287279 = 410810399
410557963 = (1637 * 250799) and 410557963 + 1637 + 250799 = 410810399
410597779 = (1949 * 210671) and 410597779 + 1949 + 210671 = 410810399
410600941 = (1979 * 207479) and 410600941 + 1979 + 207479 = 410810399
410628563 = (2287 * 179549) and 410628563 + 2287 + 179549 = 410810399
410630251 = (2309 * 177839) and 410630251 + 2309 + 177839 = 410810399
410653301 = (2659 * 154439) and 410653301 + 2659 + 154439 = 410810399
410668837 = (2963 * 138599) and 410668837 + 2963 + 138599 = 410810399
410669111 = (2969 * 138319) and 410669111 + 2969 + 138319 = 410810399
410673941 = (3079 * 133379) and 410673941 + 3079 + 133379 = 410810399
410684207 = (3343 * 122849) and 410684207 + 3343 + 122849 = 410810399
410688143 = (3457 * 118799) and 410688143 + 3457 + 118799 = 410810399
410697439 = (3761 * 109199) and 410697439 + 3761 + 109199 = 410810399
410723851 = (5039 * 81509) and 410723851 + 5039 + 81509 = 410810399
410734327 = (5849 * 70223) and 410734327 + 5849 + 70223 = 410810399
410734349 = (5851 * 70199) and 410734349 + 5851 + 70199 = 410810399
410738611 = (6269 * 65519) and 410738611 + 6269 + 65519 = 410810399
410750611 = (7919 * 51869) and 410750611 + 7919 + 51869 = 410810399
410759351 = (10009 * 41039) and 410759351 + 10009 + 41039 = 410810399
410760101 = (10259 * 40039) and 410760101 + 10259 + 40039 = 410810399
410761151 = (10639 * 38609) and 410761151 + 10639 + 38609 = 410810399
410761733 = (10867 * 37799) and 410761733 + 10867 + 37799 = 410810399
410762263 = (11087 * 37049) and 410762263 + 11087 + 37049 = 410810399
410763589 = (11699 * 35111) and 410763589 + 11699 + 35111 = 410810399
410764111 = (11969 * 34319) and 410764111 + 11969 + 34319 = 410810399
410766691 = (13679 * 30029) and 410766691 + 13679 + 30029 = 410810399
410769277 = (17099 * 24023) and 410769277 + 17099 + 24023 = 410810399
410769517 = (17783 * 23099) and 410769517 + 17783 + 23099 = 410810399
410769629 = (18199 * 22571) and 410769629 + 18199 + 22571 = 410810399
410769851 = (19759 * 20789) and 410769851 + 19759 + 20789 = 410810399

63 [643242593, 673873189, 704503777, 710629891, 718426757, 726382717, 728065697, 729565069, 730758433, 731050141, 731458529, 731852327, 732071101, 732216949, 732349537, 732777889, 733092001, 733383661, 733675297, 733692451, 733848629, 733908577, 734086499, 734112661, 734178911, 734276743, 734282687, 734301917, 734433223, 734460109, 734514413, 734618891, 734734753, 734739829, 734789059, 734804557, 734817901, 734831653, 734861411, 734883911, 734899183, 734936141, 734950141, 734974949, 734975201, 734986453, 734990497, 735011413, 735015649, 735020437, 735031229, 735045601, 735046427, 735055051, 735059203, 735065197, 735065701, 735071357, 735074401, 735077153, 735078661, 735078767, 735080173] -> 735134399

643242593 = (7 * 91891799) and 643242593 + 7 + 91891799 = 735134399
673873189 = (11 * 61261199) and 673873189 + 11 + 61261199 = 735134399
704503777 = (23 * 30630599) and 704503777 + 23 + 30630599 = 735134399
710629891 = (29 * 24504479) and 710629891 + 29 + 24504479 = 735134399
718426757 = (43 * 16707599) and 718426757 + 43 + 16707599 = 735134399
726382717 = (83 * 8751599) and 726382717 + 83 + 8751599 = 735134399
728065697 = (103 * 7068599) and 728065697 + 103 + 7068599 = 735134399
729565069 = (131 * 5569199) and 729565069 + 131 + 5569199 = 735134399
730758433 = (167 * 4375799) and 730758433 + 167 + 4375799 = 735134399
731050141 = (179 * 4084079) and 731050141 + 179 + 4084079 = 735134399
731458529 = (199 * 3675671) and 731458529 + 199 + 3675671 = 735134399
731852327 = (223 * 3281849) and 731852327 + 223 + 3281849 = 735134399
732071101 = (239 * 3063059) and 732071101 + 239 + 3063059 = 735134399
732216949 = (251 * 2917199) and 732216949 + 251 + 2917199 = 735134399
732349537 = (263 * 2784599) and 732349537 + 263 + 2784599 = 735134399
732777889 = (311 * 2356199) and 732777889 + 311 + 2356199 = 735134399
733092001 = (359 * 2042039) and 733092001 + 359 + 2042039 = 735134399
733383661 = (419 * 1750319) and 733383661 + 419 + 1750319 = 735134399
733675297 = (503 * 1458599) and 733675297 + 503 + 1458599 = 735134399
733692451 = (509 * 1441439) and 733692451 + 509 + 1441439 = 735134399
733848629 = (571 * 1285199) and 733848629 + 571 + 1285199 = 735134399
733908577 = (599 * 1225223) and 733908577 + 599 + 1225223 = 735134399
734086499 = (701 * 1047199) and 734086499 + 701 + 1047199 = 735134399
734112661 = (719 * 1021019) and 734112661 + 719 + 1021019 = 735134399
734178911 = (769 * 954719) and 734178911 + 769 + 954719 = 735134399
734276743 = (857 * 856799) and 734276743 + 857 + 856799 = 735134399
734282687 = (863 * 850849) and 734282687 + 863 + 850849 = 735134399
734301917 = (883 * 831599) and 734301917 + 883 + 831599 = 735134399
734433223 = (1049 * 700127) and 734433223 + 1049 + 700127 = 735134399
734460109 = (1091 * 673199) and 734460109 + 1091 + 673199 = 735134399
734514413 = (1187 * 618799) and 734514413 + 1187 + 618799 = 735134399
734618891 = (1429 * 514079) and 734618891 + 1429 + 514079 = 735134399
734734753 = (1847 * 397799) and 734734753 + 1847 + 397799 = 735134399
734739829 = (1871 * 392699) and 734739829 + 1871 + 392699 = 735134399
734789059 = (2141 * 343199) and 734789059 + 2141 + 343199 = 735134399
734804557 = (2243 * 327599) and 734804557 + 2243 + 327599 = 735134399
734817901 = (2339 * 314159) and 734817901 + 2339 + 314159 = 735134399
734831653 = (2447 * 300299) and 734831653 + 2447 + 300299 = 735134399
734861411 = (2719 * 270269) and 734861411 + 2719 + 270269 = 735134399
734883911 = (2969 * 247519) and 734883911 + 2969 + 247519 = 735134399
734899183 = (3167 * 232049) and 734899183 + 3167 + 232049 = 735134399
734936141 = (3779 * 194479) and 734936141 + 3779 + 194479 = 735134399
734950141 = (4079 * 180179) and 734950141 + 4079 + 180179 = 735134399
734974949 = (4751 * 154699) and 734974949 + 4751 + 154699 = 735134399
734975201 = (4759 * 154439) and 734975201 + 4759 + 154439 = 735134399
734986453 = (5147 * 142799) and 734986453 + 5147 + 142799 = 735134399
734990497 = (5303 * 138599) and 734990497 + 5303 + 138599 = 735134399
735011413 = (6299 * 116687) and 735011413 + 6299 + 116687 = 735134399
735015649 = (6551 * 112199) and 735015649 + 6551 + 112199 = 735134399
735020437 = (6863 * 107099) and 735020437 + 6863 + 107099 = 735134399
735031229 = (7699 * 95471) and 735031229 + 7699 + 95471 = 735134399
735045601 = (9239 * 79559) and 735045601 + 9239 + 79559 = 735134399
735046427 = (9349 * 78623) and 735046427 + 9349 + 78623 = 735134399
735055051 = (10709 * 68639) and 735055051 + 10709 + 68639 = 735134399
735059203 = (11549 * 63647) and 735059203 + 11549 + 63647 = 735134399
735065197 = (13103 * 56099) and 735065197 + 13103 + 56099 = 735134399
735065701 = (13259 * 55439) and 735065701 + 13259 + 55439 = 735134399
735071357 = (15443 * 47599) and 735071357 + 15443 + 47599 = 735134399
735074401 = (17159 * 42839) and 735074401 + 17159 + 42839 = 735134399
735077153 = (19447 * 37799) and 735077153 + 19447 + 37799 = 735134399
735078661 = (21419 * 34319) and 735078661 + 21419 + 34319 = 735134399
735078767 = (21599 * 34033) and 735078767 + 21599 + 34033 = 735134399
735080173 = (26927 * 27299) and 735080173 + 26927 + 27299 = 735134399

***

Gennady wrote:

Q1: the larger example for chain of 2 consequtive integers:

368428355231=605879*608089 and 368429569199=605879+608089+368428355231

368428355233=606899*607067 and 368429569199=606899+607067+368428355233

And if not consequtive ones:

n= 2

994054224341=993479*1000579 and 994056218399=993479+1000579+994054224341

994054224347=994453*999599 and 994056218399=994453+999599+994054224347

n= 3

1012213282393=993887*1018439 and 1012215294719=993887+1018439+1012213282393

1012213282489=998759*1013471 and 1012215294719=998759+1013471+1012213282489

1012213282537=1003679*1008503 and 1012215294719=1003679+1008503+1012213282537

Q2: The list of chain of length from 2 to 21 (the smallest integers of every n).

Exeptionally exactly equal to n (for examle for n=7 the integers less than for n=5 and 6)

n= 2

33=3*11 and 47=3+11+33

35=5*7 and 47=5+7+35

n= 3

141=3*47 and 191=3+47+141

155=5*31 and 191=5+31+155

161=7*23 and 191=7+23+161

n= 4

267=3*89 and 359=3+89+267

295=5*59 and 359=5+59+295

319=11*29 and 359=11+29+319

323=17*19 and 359=17+19+323

n= 5

2643=3*881 and 3527=3+881+2643

2935=5*587 and 3527=5+587+2935

3223=11*293 and 3527=11+293+3223

3263=13*251 and 3527=13+251+3263

3403=41*83 and 3527=41+83+3403

n= 6

2157=3*719 and 2879=3+719+2157

2395=5*479 and 2879=5+479+2395

2513=7*359 and 2879=7+359+2513

2629=11*239 and 2879=11+239+2629

2759=31*89 and 2879=31+89+2759

2773=47*59 and 2879=47+59+2773

n= 7

1077=3*359 and 1439=3+359+1077

1195=5*239 and 1439=5+239+1195

1253=7*179 and 1439=7+179+1253

1343=17*79 and 1439=17+79+1343

1349=19*71 and 1439=19+71+1349

1357=23*59 and 1439=23+59+1357

1363=29*47 and 1439=29+47+1363

n= 8

5937=3*1979 and 7919=3+1979+5937

6595=5*1319 and 7919=5+1319+6595

7249=11*659 and 7919=11+659+7249

7463=17*439 and 7919=17+439+7463

7627=29*263 and 7919=29+263+7627

7697=43*179 and 7919=43+179+7697

7729=59*131 and 7919=59+131+7729

7739=71*109 and 7919=71+109+7739

n= 9

14037=3*4679 and 18719=3+4679+14037

15595=5*3119 and 18719=5+3119+15595

16373=7*2339 and 18719=7+2339+16373

17149=11*1559 and 18719=11+1559+17149

17663=17*1039 and 18719=17+1039+17663

18283=47*389 and 18719=47+389+18283

18349=59*311 and 18719=59+311+18349

18407=79*233 and 18719=79+233+18407

18437=103*179 and 18719=103+179+18437

n=10

31495=5*6299 and 37799=5+6299+31495

35087=13*2699 and 37799=13+2699+35087

35683=17*2099 and 37799=17+2099+35683

35891=19*1889 and 37799=19+1889+35891

36511=29*1259 and 37799=29+1259+36511

37267=83*449 and 37799=83+449+37267

37291=89*419 and 37799=89+419+37291

37343=107*349 and 37799=107+349+37343

37391=139*269 and 37799=139+269+37391

37399=149*251 and 37799=149+251+37399

n=11

95741=19*5039 and 100799=19+5039+95741

97411=29*3359 and 100799=29+3359+97411

98359=41*2399 and 100799=41+2399+98359

98653=47*2099 and 100799=47+2099+98653

99329=71*1399 and 100799=71+1399+99329

99461=79*1259 and 100799=79+1259+99461

99941=139*719 and 100799=139+719+99941

100033=167*599 and 100799=167+599+100033

100097=199*503 and 100799=199+503+100097

100127=223*449 and 100799=223+449+100127

100141=239*419 and 100799=239+419+100141

n=12

41577=3*13859 and 55439=3+13859+41577

46195=5*9239 and 55439=5+9239+46195

52343=17*3079 and 55439=17+3079+52343

53107=23*2309 and 55439=23+2309+53107

53563=29*1847 and 55439=29+1847+53563

54079=41*1319 and 55439=41+1319+54079

54137=43*1259 and 55439=43+1259+54137

54599=71*769 and 55439=71+769+54599

54697=83*659 and 55439=83+659+54697

54827=109*503 and 55439=109+503+54827

54889=131*419 and 55439=131+419+54889

54953=179*307 and 55439=179+307+54953

n=13

97013=7*13859 and 110879=7+13859+97013

101629=11*9239 and 110879=11+9239+101629

102947=13*7919 and 110879=13+7919+102947

108523=47*2309 and 110879=47+2309+108523

108973=59*1847 and 110879=59+1847+108973

109477=83*1319 and 110879=83+1319+109477

109559=89*1231 and 110879=89+1231+109559

109909=131*839 and 110879=131+839+109909

110053=167*659 and 110879=167+659+110053

110179=239*461 and 110879=239+461+110179

110189=251*439 and 110879=251+439+110189

110197=263*419 and 110879=263+419+110197

110213=307*359 and 110879=307+359+110213

n=14

158753=7*22679 and 181439=7+22679+158753

168467=13*12959 and 181439=13+12959+168467

171343=17*10079 and 181439=17+10079+171343

173857=23*7559 and 181439=23+7559+173857

175363=29*6047 and 181439=29+6047+175363

175739=31*5669 and 181439=31+5669+175739

177613=47*3779 and 181439=47+3779+177613

178027=53*3359 and 181439=53+3359+178027

178357=59*3023 and 181439=59+3023+178357

179093=79*2267 and 181439=79+2267+179093

180407=223*809 and 181439=223+809+180407

180469=251*719 and 181439=251+719+180469

180577=359*503 and 181439=359+503+180577

180589=419*431 and 181439=419+431+180589

n=15

124737=3*41579 and 166319=3+41579+124737

145523=7*20789 and 166319=7+20789+145523

152449=11*13859 and 166319=11+13859+152449

157063=17*9239 and 166319=17+9239+157063

162497=43*3779 and 166319=43+3779+162497

163187=53*3079 and 166319=53+3079+163187

163939=71*2309 and 166319=71+2309+163939

164257=83*1979 and 166319=83+1979+164257

164383=89*1847 and 166319=89+1847+164383

164699=109*1511 and 166319=109+1511+164699

164929=131*1259 and 166319=131+1259+164929

164993=139*1187 and 166319=139+1187+164993

165283=197*839 and 166319=197+839+165283

165409=251*659 and 166319=251+659+165409

165499=359*461 and 166319=359+461+165499

n=16

644459=31*20789 and 665279=31+20789+644459

651373=47*13859 and 665279=47+13859+651373

654133=59*11087 and 665279=59+11087+654133

655969=71*9239 and 665279=71+9239+655969

657277=83*7919 and 665279=83+7919+657277

659123=109*6047 and 665279=109+6047+659123

660109=131*5039 and 665279=131+5039+660109

660389=139*4751 and 665279=139+4751+660389

661723=197*3359 and 665279=197+3359+661723

662087=223*2969 and 665279=223+2969+662087

663073=359*1847 and 665279=359+1847+663073

663277=419*1583 and 665279=419+1583+663277

663329=439*1511 and 665279=439+1511+663329

663379=461*1439 and 665279=461+1439+663379

663457=503*1319 and 665279=503+1319+663457

663647=769*863 and 665279=769+863+663647

n=17

471229=11*42839 and 514079=11+42839+471229

485503=17*28559 and 514079=17+28559+485503

488357=19*25703 and 514079=19+25703+488357

492637=23*21419 and 514079=23+21419+492637

501799=41*12239 and 514079=41+12239+501799

503323=47*10709 and 514079=47+10709+503323

506453=67*7559 and 514079=67+7559+506453

508279=89*5711 and 514079=89+5711+508279

508939=101*5039 and 514079=101+5039+508939

509213=107*4759 and 514079=107+4759+509213

510269=139*3671 and 514079=139+3671+510269

511699=239*2141 and 514079=239+2141+511699

511789=251*2039 and 514079=251+2039+511789

511919=271*1889 and 514079=271+1889+511919

512293=359*1427 and 514079=359+1427+512293

512437=419*1223 and 514079=419+1223+512437

512557=503*1019 and 514079=503+1019+512557

n=18

6278429=151*41579 and 6320159=151+41579+6278429

6282373=167*37619 and 6320159=167+37619+6282373

6284869=179*35111 and 6320159=179+35111+6284869

6303149=379*16631 and 6320159=379+16631+6303149

6305099=431*14629 and 6320159=431+14629+6305099

6306019=461*13679 and 6320159=461+13679+6306019

6307117=503*12539 and 6320159=503+12539+6307117

6308503=569*11087 and 6320159=569+11087+6308503

6308927=593*10639 and 6320159=593+10639+6308927

6310237=683*9239 and 6320159=683+9239+6310237

6311443=797*7919 and 6320159=797+7919+6311443

6311797=839*7523 and 6320159=839+7523+6311797

6313157=1063*5939 and 6320159=1063+5939+6313157

6314053=1319*4787 and 6320159=1319+4787+6314053

6314587=1583*3989 and 6320159=1583+3989+6314587

6314927=1889*3343 and 6320159=1889+3343+6314927

6314989=1979*3191 and 6320159=1979+3191+6314989

6315047=2089*3023 and 6320159=2089+3023+6315047

n=19

869377=23*37799 and 907199=23+37799+869377

878819=31*28349 and 907199=31+28349+878819

885559=41*21599 and 907199=41+21599+885559

888253=47*18899 and 907199=47+18899+888253

896317=83*10799 and 907199=83+10799+896317

897031=89*10079 and 907199=89+10079+897031

901003=149*6047 and 907199=149+6047+901003

901633=167*5399 and 907199=167+5399+901633

901981=179*5039 and 907199=179+5039+901981

902927=223*4049 and 907199=223+4049+902927

903181=239*3779 and 907199=239+3779+903181

903571=269*3359 and 907199=269+3359+903571

904259=349*2591 and 907199=349+2591+904259

904669=431*2099 and 907199=431+2099+904669

904831=479*1889 and 907199=479+1889+904831

905089=599*1511 and 907199=599+1511+905089

905153=647*1399 and 907199=647+1399+905153

905221=719*1259 and 907199=719+1259+905221

905287=863*1049 and 907199=863+1049+905287

n=20

790001=19*41579 and 831599=19+41579+790001

796927=23*34649 and 831599=23+34649+796927

812657=43*18899 and 831599=43+18899+812657

817681=59*13859 and 831599=59+13859+817681

819979=71*11549 and 831599=71+11549+819979

822271=89*9239 and 831599=89+9239+822271

823793=107*7699 and 831599=107+7699+823793

823931=109*7559 and 831599=109+7559+823931

825169=131*6299 and 831599=131+6299+825169

825521=139*5939 and 831599=139+5939+825521

827243=199*4157 and 831599=199+4157+827243

828049=251*3299 and 831599=251+3299+828049

828251=269*3079 and 831599=269+3079+828251

828593=307*2699 and 831599=307+2699+828593

828931=359*2309 and 831599=359+2309+828931

829201=419*1979 and 831599=419+1979+829201

829271=439*1889 and 831599=439+1889+829271

829303=449*1847 and 831599=449+1847+829303

829607=593*1399 and 831599=593+1399+829607

829681=659*1259 and 831599=659+1259+829681

n=21

17421601=419*41579 and 17463599=419+41579+17421601

17425339=461*37799 and 17463599=461+37799+17425339

17428447=503*34649 and 17463599=503+34649+17428447

17433607=593*29399 and 17463599=593+29399+17433607

17436481=659*26459 and 17463599=659+26459+17436481

17440151=769*22679 and 17463599=769+22679+17440151

17441231=809*21559 and 17463599=809+21559+17441231

17441971=839*20789 and 17463599=839+20789+17441971

17445919=1049*16631 and 17463599=1049+16631+17445919

17447713=1187*14699 and 17463599=1187+14699+17447713

17448481=1259*13859 and 17463599=1259+13859+17448481

17449051=1319*13229 and 17463599=1319+13229+17449051

17449727=1399*12473 and 17463599=1399+12473+17449727

17450539=1511*11549 and 17463599=1511+11549+17450539

17452471=1889*9239 and 17463599=1889+9239+17452471

17452801=1979*8819 and 17463599=1979+8819+17452801

17453633=2267*7699 and 17463599=2267+7699+17453633

17453731=2309*7559 and 17463599=2309+7559+17453731

17454721=2939*5939 and 17463599=2939+5939+17454721

17454751=2969*5879 and 17463599=2969+5879+17454751

17454851=3079*5669 and 17463599=3079+5669+17454851

if integers of different parity are allowed then there ara chains with smaller integers

n= 2

14=2*7 and 23=2+7+14

15=3*5 and 23=3+5+15

n= 3

46=2*23 and 71=2+23+46

51=3*17 and 71=3+17+51

55=5*11 and 71=5+11+55

n= 4

158=2*79 and 239=2+79+158

177=3*59 and 239=3+59+177

203=7*29 and 239=7+29+203

209=11*19 and 239=11+19+209

n= 5

478=2*239 and 719=2+239+478

537=3*179 and 719=3+179+537

623=7*89 and 719=7+89+623

649=11*59 and 719=11+59+649

667=23*29 and 719=23+29+667

n= 8

958=2*479 and 1439=2+479+958

1077=3*359 and 1439=3+359+1077

1195=5*239 and 1439=5+239+1195

1253=7*179 and 1439=7+179+1253

1343=17*79 and 1439=17+79+1343

1349=19*71 and 1439=19+71+1349

1357=23*59 and 1439=23+59+1357

1363=29*47 and 1439=29+47+1363

n= 9

6718=2*3359 and 10079=2+3359+6718

8813=7*1259 and 10079=7+1259+8813

9229=11*839 and 10079=11+839+9229

9347=13*719 and 10079=13+719+9347

9557=19*503 and 10079=19+503+9557

9637=23*419 and 10079=23+419+9637

9799=41*239 and 10079=41+239+9799

9853=59*167 and 10079=59+167+9853

9869=71*139 and 10079=71+139+9869

***

Paul wrote:

Q1.

Here are my solutions when there are consecutive same parity integers.

33=3*11 and 33+3+11=47

35=5*7 and 35+5+7=47

25019=127*197 and 25019+127+197=25343

25021=131*191 and 25021+131+191=25343

205727=409*503 and 205727+409+503=206639

205729=419*491 and 205729+419+491=206639

1887239=1249*1511 and 1887239+1249+1511=1889999

1887241=1259*1499 and 1887241+1259+1499=1889999

4632491=2069*2239 and 4632491+2069+2239=4636799

4632493=2099*2207 and 4632493+2099+2207=4636799

28475387=5023*5669 and 28475387+5023+5669=28486079

28475389=5039*5651 and 28475389+5039+5651=28486079

173112767=12473*13879 and 173112767+12473+13879=173139119

173112769=12491*13859 and 173112769+12491+13859=173139119

184865201=13339*13859 and 184865201+13339+13859=184892399

184865203=13397*13799 and 184865203+13397+13799=184892399

469997117=21419*21943 and 469997117+21419+21943=470040479

469997119=21521*21839 and 469997119+21521+21839=470040479

488936561=21559*22679 and 488936561+21559+22679=488980799

488936563=21599*22637 and 488936563+21599+22637=488980799

513124019=21997*23327 and 513124019+21997+23327=513169343

513124021=22031*23291 and 513124021+22031+23291=513169343

1182466421=34019*34759 and 1182466421+34019+34759=1182535199

1182466423=34127*34649 and 1182466423+34127+34649=1182535199

2311550237=47599*48563 and 2311550237+47599+48563=2311646399

2311550239=47711*48449 and 2311550239+47711+48449=2311646399

2336270927=48023*48649 and 2336270927+48023+48649=2336367599

2336270929=48299*48371 and 2336270929+48299+48371=2336367599

2498177621=49139*50839 and 2498177621+49139+50839=2498277599

2498177623=49199*50777 and 2498177623+49199+50777=2498277599

2612240597=49363*52919 and 2612240597+49363+52919=2612342879

2612240599=49391*52889 and 2612240599+49391+52889=2612342879

2652549791=51169*51839 and 2652549791+51169+51839=2652652799

2652549793=51407*51599 and 2652549793+51407+51599=2652652799

2864243819=51407*55717 and 2864243819+51407+55717=2864350943

2864243821=51431*55691 and 2864243821+51431+55691=2864350943

3224831189=55399*58211 and 3224831189+55399+58211=3224944799

3224831191=55439*58169 and 3224831191+55439+58169=3224944799

3787648517=59443*63719 and 3787648517+59443+63719=3787771679

3787648519=59471*63689 and 3787648519+59471+63689=3787771679

3942868001=61879*63719 and 3942868001+61879+63719=3942993599

3942868003=61949*63647 and 3942868003+61949+63647=3942993599

4115362829=62099*66271 and 4115362829+62099+66271=4115491199

4115362831=62129*66239 and 4115362831+62129+66239=4115491199

4516340627=66653*67759 and 4516340627+66653+67759=4516475039

4516340629=66791*67619 and 4516340629+66791+67619=4516475039

4562098031=65239*69929 and 4562098031+65239+69929=4562233199

4562098033=65267*69899 and 4562098033+65267+69899=4562233199

4825615127=63409*76103 and 4825615127+63409+76103=4825754639

4825615129=63419*76091 and 4825615129+63419+76091=4825754639

5336125007=68863*77489 and 5336125007+68863+77489=5336271359

5336125009=68879*77471 and 5336125009+68879+77471=5336271359

5510206619=60607*90917 and 5510206619+60607+90917=5510358143

5510206621=60611*90911 and 5510206621+60611+90911=5510358143

6622843769=80387*82387 and 6622843769+80387+82387=6623006543

6622843771=80471*82301 and 6622843771+80471+82301=6623006543

7271641541=77839*93419 and 7271641541+77839+93419=7271812799

7271641543=77849*93407 and 7271641543+77849+93407=7271812799

7983239597=81559*97883 and 7983239597+81559+97883=7983419039

7983239599=81569*97871 and 7983239599+81569+97871=7983419039

If we now relax the consecutive part then we can find many more solutions with many solutions per same prime result. Here is one with 45 solutions.

No of terms = 45 with the prime 2049465599

1793282393 = 7*256183199 and 1793282393+7+256183199=2049465599

1935606383 = 17*113859199 and 1935606383+17+113859199=2049465599

2015307781 = 59*34157759 and 2015307781+59+34157759=2049465599

2019326333 = 67*30139199 and 2019326333+67+30139199=2049465599

2025067117 = 83*24398399 and 2025067117+83+24398399=2049465599

2033454023 = 127*16011449 and 2033454023+127+16011449=2049465599

2034826421 = 139*14639039 and 2034826421+139+14639039=2049465599

2037266233 = 167*12199199 and 2037266233+167+12199199=2049465599

2038791109 = 191*10674299 and 2038791109+191+10674299=2049465599

2040554651 = 229*8910719 and 2040554651+229+8910719=2049465599

2041930529 = 271*7534799 and 2041930529+271+7534799=2049465599

2042896489 = 311*6568799 and 2042896489+311+6568799=2049465599

2043896033 = 367*5569199 and 2043896033+367+5569199=2049465599

2046618401 = 719*2846479 and 2046618401+719+2846479=2049465599

2046989573 = 827*2475199 and 2046989573+827+2475199=2049465599

2047024921 = 839*2439839 and 2047024921+839+2439839=2049465599

2047512679 = 1049*1951871 and 2047512679+1049+1951871=2049465599

2047580813 = 1087*1883699 and 2047580813+1087+1883699=2049465599

2048412643 = 1949*1051007 and 2048412643+1949+1051007=2049465599

2048712151 = 2729*750719 and 2048712151+2729+750719=2049465599

2048868103 = 3449*594047 and 2048868103+3449+594047=2049465599

2048968781 = 4159*492659 and 2048968781+4159+492659=2049465599

2048982917 = 4283*478399 and 2048982917+4283+478399=2049465599

2049010619 = 4549*450431 and 2049010619+4549+450431=2049465599

2049033829 = 4799*426971 and 2049033829+4799+426971=2049465599

2049146249 = 6551*312799 and 2049146249+6551+312799=2049465599

2049153901 = 6719*304979 and 2049153901+6719+304979=2049465599

2049161677 = 6899*297023 and 2049161677+6899+297023=2049465599

2049224719 = 8831*232049 and 2049224719+8831+232049=2049465599

2049224921 = 8839*231839 and 2049224921+8839+231839=2049465599

2049228151 = 8969*228479 and 2049228151+8969+228479=2049465599

2049246509 = 9791*209299 and 2049246509+9791+209299=2049465599

2049256397 = 10303*198899 and 2049256397+10303+198899=2049465599

2049258137 = 10399*197063 and 2049258137+10399+197063=2049465599

2049306557 = 14143*144899 and 2049306557+14143+144899=2049465599

2049309671 = 14489*141439 and 2049309671+14489+141439=2049465599

2049324379 = 16421*124799 and 2049324379+16421+124799=2049465599

2049346751 = 20929*97919 and 2049346751+20929+97919=2049465599

2049354617 = 23399*87583 and 2049354617+23399+87583=2049465599

2049354781 = 23459*87359 and 2049354781+23459+87359=2049465599

2049360281 = 25759*79559 and 2049360281+25759+79559=2049465599

2049365497 = 28703*71399 and 2049365497+28703+71399=2049465599

2049370601 = 33119*61879 and 2049370601+33119+61879=2049465599

2049372601 = 35879*57119 and 2049372601+35879+57119=2049465599

2049374603 = 40949*50047 and 2049374603+40949+50047=2049465599

My method for finding these solutions is as follows using:-

A * B + C = Pt

The target Prime is Pt, we then list all primes up to the Sqr root of Pt, set these as Pi. Then,

A = Pi

B = (Pt - Pi)/(Pi + 1). Test if this is a prime if so we have a solution Then.

C = ((Pt - Pi)*Pi)/ (Pi + 1).

***

Giorgos wrote:
Q1: If we are searching for pairs of numbers with the same parity with any number of prime factors,
then the possible primes are:

47, 71, 89, 167, 239, 251, 331, 383, 419, 439, 491, 503, 503, 571, 587, 599, 599, 647, 659, 683, 683, 739, 787, 829, 1049, 1151, 1223, 1259, 1289, 1301, 1319, 1319, 1367, 1699...

e.g. the last number 1699 produces the following pair

1602 = 2*3*3*89 and 1602 + 2 + 3 + 3 + 89 = 1699

1666 = 2*7*7*17 and 1666 + 2 + 7 + 7 + 17 = 1699

If we don't care about parity, there are more ways to express these primes. For example, in Bergot's example we can also have

254 = 2*127 and 254 + 2 + 127 = 383

If we are searching for pairs of numbers with the same parity with exactly 2 prime factors (Semiprimes), then the possible primes are:
47, 71, 167, 251, 383, 503, 587, 599, 647, 683, 1151, 1223, 1259, 1319, 1367, 2099...

The largest prime that I could find that can be expressed in exactly 2 ways is 10^60+3439 :

999985469340308049985469340308049985469340308049985469274929 = 68819 * 14530659691950014530659691950014530659691950014530659691 and

999985469340308049985469340308049985469340308049985469274929 + 68819 + 14530659691950014530659691950014530659691950014530659691 = 10^60+3439

999999999999998814302835590138818580525657485842502606958483 = 843385672173479 * 1185697164409861181419474342513314111720871477 and

999999999999998814302835590138818580525657485842502606958483 + 843385672173479 + 1185697164409861181419474342513314111720871477 = 10^60+3439

It is pretty easy to find even bigger primes.

Q2. It is possible to find more than two numbers with the same parity that return the same prime result.

Here is a list of primes that can be expressed in k number of ways:

k number of ways | primes P

_______________|_____________________

3 -> 239
4 -> 359
7 -> 1439
8 -> 7919
9 -> 18719

10 -> 37799
12 -> 55439
13 -> 110879
15 -> 166319
18 -> 514079
19 -> 665279
23 -> 831599

. . . . . . . . . . . . . . . . . . .

168 -> 30561299999999999999999999
186 -> 52598699999999999999999999
199 -> 63617399999999999999999999
221 -> 79361099999999999999999999

and the list goes on...

One observation that helped me to easily find these primes is that their digitsum mod 9 is congruent to 8.

All these numbers that produce those primes in my examples are semiprimes.

So, the prime 79361099999999999999999999 can be expressed in 221 ways.

I'm sending you a txt file with all 221 numbers in the same format as the example of Bergot,

It is also easy to find bigger primes that can be expressed in more ways.

***

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