Problems & Puzzles: Puzzles

Puzzle 670 The same sum of divisors Shyam Sunder Gupta asks  in his CYF NO. 29  for three consecutive numbers whose sum of divisors is same. Mukesh Kr Jha dt comments at 23-07-2012: No solution found for this problem for n from 1 to 3,00,00,000 [sic] using Maple application. J. K. Andersen comments at 28-12-12: http://oeis.org/A002961 shows cases of two consecutive numbers. The table by T. D. Noe of terms up to 10^12 has no cases of three consecutive numbers. Carlos Rivera has counted the 4804 cases listed by T. D. Noe, grouped by its last digit hoping to find a clue to speed up this search. Last digit Frequency 0 71 1 369 2 383 3 395 4 1206 5 1229 6 352 7 357 8 374 9 68  Total 4804  Q. Send the results of your search.

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Contributions came from Jan van Delden

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Jan wrote:

Factoring numbers of this size requires a considerable time.

I constructed a large sieve to this purpose, but that doesn’t seem to help a lot.

The best I can do (without other tricks) is extend the given list by a few numbers:

 

Number: 1000001331209 #Divisors: 4 Divisors: 19 1999 3511 7499 sum: 13028

Number: 1000001331210 #Divisors: 7 Divisors: 2 3 5 7 127 4441 8443 sum: 13028

Number: 1000006403131 #Divisors: 3 Divisors: 113 15467 572161 sum: 587741

Number: 1000006403132 #Divisors: 5 Divisors: 2 2 11 41617 546109 sum: 587741

Number: 1000009265660 #Divisors: 6 Divisors: 2 2 5 251 6827 29179 sum: 36266

Number: 1000009265661 #Divisors: 6 Divisors: 3 3 7 271 1709 34273 sum: 36266

Number: 1000010921006 #Divisors: 5 Divisors: 2 107 281 1913 8693 sum: 10996

Number: 1000010921007 #Divisors: 6 Divisors: 3 7 7 239 4759 5981 sum: 10996

Number: 1000011047139 #Divisors: 7 Divisors: 3 3 13 13 67 211 46507 sum: 46817

Number: 1000011047140 #Divisors: 7 Divisors: 2 2 5 7 23 8011 38767 sum: 46817

Number: 1000018797129 #Divisors: 7 Divisors: 3 3 3 11 41 4517 18181 sum: 22759

Number: 1000018797130 #Divisors: 6 Divisors: 2 5 19 181 1373 21179 sum: 22759

Number: 1000019050750 #Divisors: 7 Divisors: 2 5 5 5 97 1861 22159 sum: 24134

Number: 1000019050751 #Divisors: 4 Divisors: 7 1117 9391 13619 sum: 24134

Number: 1000021212575 #Divisors: 8 Divisors: 5 5 7 7 11 181 263 1559 sum: 2038

Number: 1000021212576 #Divisors: 13 Divisors: 2 2 2 2 2 3 3 3 3 3 199 487 1327 sum: 2038

Number: 1000022168025 #Divisors: 7 Divisors: 3 3 3 5 5 3847 385109 sum: 388975

Number: 1000022168026 #Divisors: 6 Divisors: 2 7 7 41 641 388277 sum: 388975

Number: 1000022709374 #Divisors: 6 Divisors: 2 13 83 127 1427 2557 sum: 4209

Number: 1000022709375 #Divisors: 11 Divisors: 3 3 3 3 5 5 5 5 5 1453 2719 sum: 4209

Number: 1000023240048 #Divisors: 10 Divisors: 2 2 2 2 3 7 7 229 587 3163 sum: 4004

Number: 1000023240049 #Divisors: 4 Divisors: 929 1013 1013 1049 sum: 4004

Number: 1000023850088 #Divisors: 8 Divisors: 2 2 2 37 61 127 131 3329 sum: 3691

Number: 1000023850089 #Divisors: 7 Divisors: 3 3 11 71 167 269 3167 sum: 3691

Number: 1000025351679 #Divisors: 6 Divisors: 3 17 83 271 499 1747 sum: 2620

Number: 1000025351680 #Divisors: 13 Divisors: 2 2 2 2 2 2 2 2 2 5 347 751 1499 sum: 2620

Number: 1000028446831 #Divisors: 4 Divisors: 19 223 11867 19889 sum: 31998

Number: 1000028446832 #Divisors: 8 Divisors: 2 2 2 2 83 113 211 31583 sum: 31998

Number: 1000030941450 #Divisors: 9 Divisors: 2 3 3 5 5 23 101 163 5869 sum: 6174

Number: 1000030941451 #Divisors: 4 Divisors: 137 1499 1741 2797 sum: 6174

Number: 1000033100857 #Divisors: 6 Divisors: 7 7 7 17 8623 19889 sum: 28550

Number: 1000033100858 #Divisors: 5 Divisors: 2 11 599 3049 24889 sum: 28550

Number: 1000036424168 #Divisors: 6 Divisors: 2 2 2 37 41621 81173 sum: 122837

Number: 1000036424169 #Divisors: 5 Divisors: 3 3 47 23911 98873 sum: 122837

Number: 1000039050464 #Divisors: 9 Divisors: 2 2 2 2 2 127 359 433 1583 sum: 2512

Number: 1000039050465 #Divisors: 8 Divisors: 3 5 7 13 19 37 557 1871 sum: 2512

Number: 1000045099151 #Divisors: 6 Divisors: 7 17 37 307 601 1231 sum: 2200

Number: 1000045099152 #Divisors: 12 Divisors: 2 2 2 2 3 3 3 3 11 67 811 1291 sum: 2200

Number: 1000056935865 #Divisors: 7 Divisors: 3 5 13 89 151 263 1451 sum: 1975

Number: 1000056935866 #Divisors: 6 Divisors: 2 23 109 379 641 821 sum: 1975

Number: 1000061081071 #Divisors: 5 Divisors: 13 157 433 761 1487 sum: 2851

Number: 1000061081072 #Divisors: 9 Divisors: 2 2 2 2 17 61 73 353 2339 sum: 2851

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Fred Schneider wrote on Dec 23, 2018:

I analyzed the 10135 pairs from the B file at http://oeis.org/A002961
I found the ratio closest to 1 between the common sigma of (n and n + 1) of n-1 and n + 2.  (I took the ratio of the larger figure to the smaller for consistency in computing the ratio)  A ratio of 1 of course would indicate a sigma triple. 

sigma 2105750915125 = 3252542791680 = 2^13 * 3^8 * 5 * 7^2 * 13   * 19
sigma 2105750915127 = 3252483037440 = 2^8  * 3^3 * 5 * 7   * 13^2 * 19 * 53 * 79
(The gcd is 2^8 * 3^3 * 5 * 7 * 13 * 19 = 59754240)

For 2105750915125 and 2105750915126, the sigma ratio with 2105750915127,
1.0000183718836693, is almost 5 times as close as the next best pair.

In all, there were only four pairs that had a ratio < 1.0001.


There is a another relatively near miss 86701394, 86701395
 
sigma 86701394 = 151787520
sigma 86701396 = 151813620
The difference was only 26100 but the gcd was 2^2 * 3^2 * 5.

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First incidence of pair and closeness of ratio by magnitude of 10
first n, ratio
957, 1.1666666666666667
1634, 1.0871212121212122       (next 64665)
13088834,1.0058515080066803    (next 86701394)
86701394,1.0001719508955678    (next 9721808649, outlier more than 100 times the size)
369607342034,1.0000968625996507 (next 2105750915125)
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#d	ct	sub	odds # is 1st in pair	ratio^
2	1	1	90	-
3	2	3	450	2.0000
4	6	9	1,500	2.0000
5	15	24	6,000	1.6667
6	38	62	23,684	1.5833
7	51	113	176,471	0.8226
8	119	232	756,303	1.0531
9	301	533	2,990,033	1.2974
10	564	1097	15,957,447	1.0582
11	1198	2295	75,125,209	1.0921
12	2509	4804	358,708,649	1.0932
13	5331	10135	1,688,238,604	1.1097
Estimates
#d	ct	subtotal	odds # is 1st in pair	otr*
14	11131	21266	8,085,526,907	726,397
15	23356	44622	38,533,995,547	1,649,854
16	49008	93630	183,643,486,778	3,747,214
17	102834	196464	875,196,919,307	8,510,774

 

*otr = estimated odds of triple in digit range.  For the sake of a crude estimate, this assumes that the sigma pairs are indpendent from the sigmas of an adjacent third number.  So, the estimate that a 14-digit number has a sigma triple is approximately 1 in 726 thousand.  The estimates get progressively bleak as you can see. 
 

^rato = ratio of pairs in digit range to previous subtotal. I used the geometric mean of the ratios for digits 11-13 to get a ratio of 1.09830918 and create the estimates for digit counts 14-17.

 

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