Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1237 1089427

From the Prime Curios! pages:

The prime 1089427 divides a record 12 numbers of the form n!+1. [Merickel]

Q1. What are these twelve numbers n (please send them in a row)?

Q2. If 1089427 is the absolute champion please get the second and third champions of this process,
indicating for each one the corresponding n-values ?

 

From Set. 13-19, 2025 contrbutions came from Michael Branicky, Giorgos Kalogeropopulos, JM Rebert, Alessandro Casini, Gennady Gusev, Simon Cavegn, Emmanuel Vantieghem

***

Michael wrote:

1089427 is not the ultimate champion. These are collected in OEIS A230315 (a(n) is the smallest prime dividing n numbers of the form k! + 1).
It lists one above: 2374649, dividing 14. Two below include 62939 (9) and 292627 (11), with other numbers possibly tying these listed.

...

The 12 values for 1089427 are:
281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643, 1089426
The 14 values for 2374649 are
653598, 689779, 789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869, 2374648

***

Giorgos wrote:

Merickel has posted these records in OEIS A230315

Q1. {281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643, 1089426}

Q2. Here are all known records
2->{1}
7->{3,6}
23->{14,18,22}
59->{15,40,43,58}
71->{7,9,19,51,61,63,70}
3643->{1005,1459,1821,1858,2183,2637,2700,3642}
62939->{7669,15990,29913,31469,33025,55269,56066,56266,62938}
292627->{44009,82254,82951,106495,146313,186131,193038,209675,234482,248617,292626}
1089427->{281783,362757,439703,468331,496207,593219,621095,649723,710384,726669,807643,1089426}
2374649->{653598,689779,789933,845577,869139,946501,950026,1410794,1428147,1505509,1529071,1584715,1684869,2374648}

***

JM wrote:
Q1. I found:
1089427 divides 12 numbers of form n! + 1 with n in [281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643, 1089426].

Q2. I found:
292627 divides 11 numbers of form n! + 1 with n in [44009, 82254, 82951, 106495, 146313, 186131, 193038, 209675, 234482, 248617, 292626].
292627 divides 10 numbers of form n! + 1 with n in [44009, 82254, 82951, 106495, 146313, 186131, 193038, 209675, 234482, 248617].
62939 divides 9 numbers of form n! + 1 with n in [7669, 15990, 29913, 31469, 33025, 55269, 56066, 56.266].
3643 divides 8 numbers of form n! + 1 with n in [1005, 1459, 1821, 1858, 2183, 2637, 2700, 3642].
71 divides 7 numbers of form n! + 1 with n in [7, 9, 19, 51, 61, 63, 70].
71 divides 6 numbers of form n! + 1 with n in [7, 9, 19, 51, 61, 63].
71 divides 5 numbers of form n! + 1 with n in [7, 9, 19, 51, 61].
59 divides 4 numbers of form n! + 1 with n in [15, 40, 43, 58].
23 divides 3 numbers of form n! + 1 with n in [14, 18, 22].
7 divides 2 numbers of form n! + 1 with n in [3, 6].
2 divides 1 number of form n! + 1 with n in [1].

Later he added:

Cf. OEIS  A230315
a(n) is the smallest prime dividing n numbers of the form k! + 1.
2, 7, 23, 59, 71, 71, 71, 3643, 62939, 292627, 292627, 1089427, 2374649, 2374649
OFFSET
1,1
I found:
2374649 divides 13 numbers of form n! + 1 with n in [653598, 689779, 789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869]
2374649 divides 14 numbers of form n! + 1 with n in [653598, 689779, 789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869, 2374648]

***

Alessandro wrote_

  2. 14 - 2374649: 653598, 689779, 789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869, 2374648
    12 - 1089427: 281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643, 1089426
    11 - 292627: 44009, 82254, 82951, 106495, 146313, 186131, 193038, 209675, 234482, 248617, 292626
    10 - 480209: 47470, 78170, 128028, 134443, 158673, 321535, 323986, 345765, 478052, 480208

***

Gennady wrote:

I tested the first millions of primes. All 155 results with k>=10 are shown in the attached file all.txt
A new champion with 14 'n' was found:
p= 2374649, k=14, n=  653598    689779    789933    845577    869139    946501    950026   1410794   1428147   1505509   1529071   1584715   1684869   2374648
 
8 primes for 12 'n':
p= 1089427, k=12, n=    281783    362757    439703    468331    496207    593219    621095    649723    710384    726669    807643   1089426
p= 5975051, k=12, n=    269769    542667    924423   1172029   2987525   3136890   3795582   4803021   5050627   5432383   5705281   5975050
p= 7705331, k=12, n=    396550   1774500   2793099   3352415   3404505   4186300   4300825   4352915   4912231   5178310   7237574   7705330
p= 8294159, k=12, n=    183189    541761    689501   2092747   5000738   6201411   6826028   7333764   7604657   7752397   8110969   8294158
p= 8444981, k=12, n=     69572    690909   1034439   1213029   2058561   5291334   6386419   7093038   7231951   7410541   7754071   8444980
p=10114129, k=12, n=   1940333   2246215   2399255   4533061   4758745   5355383   5581067   7396762   7714873   7867913   8173795  10114128
p=11864791, k=12, n=    514765   2942009   3427061   4584735   5932395   7280055   8437729   8922781   9514512  10180522  11350025  11864790
p=13135877, k=12, n=    265753    402956   1104125   1197257   1209022   2866268   3880401   9255475  11938619  12031751  12870123  13135876

 

***

Simon wrote:

Q1:
1089427 divides n!+1 12 times
n:281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643, 1089426


Q2:
2374649 divides n!+1 14 times
n:653598, 689779, 789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869, 2374648

48773651 divides n!+1 13 times
n:11055031, 11829381, 13563929, 14891102, 16356385, 18166681, 24386825, 30606969, 32417265, 35209721, 36944269, 37718619, 48773650


Additional findings:
17725523 divides n!+1 14 times
n:3575753, 4441625, 4555699, 7044891, 7313131, 8862761, 10412391, 10680631, 11048500, 13169823, 13283897, 14149769, 17374384, 17725522

53016413 divides n!+1 14 times
n:2597078, 3472491, 4813131, 7028431, 7421489, 9655721, 26161977, 26854435, 43360691, 45594923, 45987981, 48203281, 49543921, 53016412

153279739 divides n!+1 14 times
n:682842, 23214559, 29044944, 33940913, 51006367, 66792321, 75509041, 76639869, 77770697, 86487417, 102273371, 119338825, 130065179, 153279738

Found no more 14 or better, searched up to prime 298750279

***

Emmanuel wrote:

The numbers  n  such that  n! + 1  is divisible by  1089427  are : 281783, 362757, 439703, 468331, 496207, 593219, 621095, 649723, 710384, 726669, 807643 and,1089426.
The next champion is the prime  p = 2374649  for which there are 14 values of  n  such that  n! + 1  is divisible by  p :  
653598, 689779, ,789933, 845577, 869139, 946501, 950026, 1410794, 1428147, 1505509, 1529071, 1584715, 1684869  and  2374649.
When I searched in the OEIS for the number  2374649  I came up to  A230315  where I learned that the next champion is > 1.1*10^8 (found by Giovanni Resta).
With the courage of despair I tried to search further from that point on.  But after two days I only reached  110050243.
I think a greater apparatus than my PC is needed to find the next champion.

***

Simon cavegn wrote on Oct 3, 2025

Q2.

I

Found a new champion!

584133523 divides n!+1 16 times
n: 52120783, 180924657, 231367367, 231615601, 259891765, 264823049, 292066761, 312913486, 319310473, 324241757, 352517921, 352766155, 403208865, 532012739, 552676296, 584133522

Later he added: "584133523 is the smallest prime for 16 times,  I tested all up to it." Accordingly this must be added in short in the OEIS A230315.

 Later on Oct 5, 2025, Simon wrote:

"it is updated: https://oeis.org/A230315"

 NB by CR: The 16 values n!+1 divided by 584133523 were verified on my request (CR) by Michael Branickly and Sabastián Martín Ruiz, independently.

***

 

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