A068383 -id:A068383 - cp's OEIS frontend

A127092 Numbers k such that k^2 divides 11^k - 1.

1, 2, 4, 5, 6, 10, 12, 20, 30, 42, 60, 84, 114, 156, 210, 222, 228, 244, 420, 444, 570, 732, 780, 798, 930, 1092, 1110, 1140, 1220, 1554, 1596, 1806, 1860, 2220, 2436, 2964, 3108, 3612, 3660, 3990, 4218, 5124, 5460, 5772, 6510, 7770, 7980, 8268, 8436, 9030

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

Subsequence of A068383.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=11 of A333500.

Cf. A068383, A127100, A127101, A127102, A127104, A127105, A127106, A127107, A292336.

Select[Range[15000], IntegerQ[(PowerMod[11, #, #^2 ]-1)/#^2 ]&]
Join[{1},Select[Range[9100],PowerMod[11,#,#^2]==1&]] (* Harvey P. Dale, Dec 30 2018 *)

for(k=1, 1e4, if(Mod(11, k^2)^k==1, print1(k", "))) \\ Seiichi Manyama, Mar 25 2020

A177911 Numbers k such that k^3 divides 11^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 20, 30, 42, 60, 68, 78, 84, 114, 122, 156, 204, 210, 222, 228, 244, 340, 366, 390, 420, 444, 546, 570, 610, 732, 780, 798, 820, 876, 930, 1010, 1020, 1092, 1110, 1140, 1164, 1218, 1220, 1428, 1482, 1554, 1596, 1806, 1830, 1860, 2020, 2220

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..1634

Cf. A068383 (k divides 11^k-1), A127092 (k^2 divides 11^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Join[{1},Select[Range[2300],PowerMod[11,#^2,#^3]==1&]] (* Harvey P. Dale, Aug 24 2024 *)

A014960 Integers n such that n divides 24^n - 1.

Original entry on oeis.org

1, 23, 529, 1081, 12167, 24863, 50807, 279841, 571849, 1168561, 2387929, 2870377, 6436343, 7009273, 13152527, 15954479, 26876903, 54922367, 66018671, 112232663, 134907719, 148035889, 161213279, 302508121, 329435831

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*24^(k-1) (cf. A014942).
All n > 1 in the sequence are multiple of 23. - Conjectured by Thomas Baruchel, Oct 10 2003; proved by Max Alekseyev, Nov 16 2019
If n is a term and prime p|(24^n - 1), then n*p is a term. In particular, if n is a term and prime p|n, then n*p is a term. The smallest term with 3 distinct prime factors is a(16) = 15954479 = 23 * 47 * 14759. - Max Alekseyev, Nov 16 2019

Max Alekseyev, Table of n, a(n) for n = 1..146

Prime factors are listed in A087807.
Cf. A014942.
Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014959 (b=22).

s = 1; Do[ If[ Mod[ s, n ] == 0, Print[n]]; s = s + (n + 1)*24^n, {n, 1, 100000}]
Join[{1},Select[Range[330*10^6],PowerMod[24,#,#]==1&]] (* Harvey P. Dale, Jan 19 2023 *)

More terms from Robert G. Wilson v, Sep 13 2000
a(9)-a(12) from Thomas Baruchel, Oct 10 2003
Edited and terms a(13) onward added by Max Alekseyev, Nov 16 2019

A014956 Positive integers k such that k divides 14^k - 1.

Original entry on oeis.org

1, 13, 169, 2041, 2197, 26533, 28561, 114413, 320437, 344929, 371293, 1487369, 4165681, 4484077, 4826809, 17962841, 19335797, 24355253, 50308609, 54153853, 58293001, 62748517, 77457601, 233516933, 249302027, 251365361, 316618289

Offset: 1

Olivier Gérard

nonn

Also, positive integers k such that k divides A014929(k).
13 divides a(n) for n > 1. All powers of 13 are terms. All a(n) that are not powers of 13 are divisible either by 157 or 677 or both. - Alexander Adamchuk, May 14 2010
Prime divisors of a(n) in order of appearance: {13, 157, 677, 11933, 122147, 52807, ...}. - Alexander Adamchuk, May 16 2010

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A128360.

Cf. A177914, A128394.

Join[{1}, Select[Range[2000000], PowerMod[14, #, #] == 1 &]] (* Robert Price, Mar 31 2020 *)

2 more terms from R. J. Mathar, Mar 05 2008
a(8)-a(23) from Alexander Adamchuk, May 14 2010
a(24)-a(44) from Alexander Adamchuk, May 16 2010
Edited by Max Alekseyev, Sep 10 2011

A014957 Positive integers k that divide 16^k - 1.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 25, 27, 39, 45, 55, 63, 75, 81, 105, 117, 125, 135, 147, 155, 165, 171, 189, 195, 205, 225, 243, 273, 275, 315, 333, 351, 375, 405, 441, 465, 495, 507, 513, 525, 567, 585, 605, 609, 615, 625, 657, 675, 729, 735, 775, 819, 825, 855, 903

Offset: 1

Olivier Gérard

nonn

Also, positive integers k that divide A014931(k).

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A014956, A177805, A177807, A128358, A128360

Cf. A128396, A177916

Join[{1},Select[Range[1000],PowerMod[16,#,#]==1&]] (* Harvey P. Dale, Jun 12 2024 *)

A014957_list = [n for n in range(1,10**6) if n == 1 or pow(16,n,n) == 1] # Chai Wah Wu, Mar 25 2021

Edited by Max Alekseyev, Sep 10 2011

A333432 A(n,k) is the n-th number m that divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 9, 8, 0, 6, 1, 5, 4, 21, 16, 0, 7, 1, 2, 25, 6, 27, 20, 0, 8, 1, 7, 3, 125, 8, 63, 32, 0, 9, 1, 2, 49, 4, 625, 12, 81, 40, 0, 10, 1, 3, 4, 343, 6, 1555, 16, 147, 64, 0, 11, 1, 2, 9, 8, 889, 8, 3125, 18, 171, 80, 0, 12

Offset: 1

Seiichi Manyama, Mar 21 2020

			Square array A(n,k) begins:
  1, 1,  1,   1,  1,     1,  1,     1,  1, ...
  2, 0,  2,   3,  2,     5,  2,     7,  2, ...
  3, 0,  4,   9,  4,    25,  3,    49,  4, ...
  4, 0,  8,  21,  6,   125,  4,   343,  8, ...
  5, 0, 16,  27,  8,   625,  6,   889, 10, ...
  6, 0, 20,  63, 12,  1555,  8,  2359, 16, ...
  7, 0, 32,  81, 16,  3125,  9,  2401, 20, ...
  8, 0, 40, 147, 18,  7775, 12,  6223, 32, ...
  9, 0, 64, 171, 24, 15625, 16, 16513, 40, ...

Seiichi Manyama, Antidiagonals n = 1..25, flattened
OEIS Wiki, 2^n mod n

Columns k=1-20 give: A000027, A063524, A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A333506, A128360.
Main diagonal gives A333433.
Cf. A333429, A333500.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=2 then `if`(n=1, 1, 0) else
        while nops(p(k)) 1 do od;
          p(k):= [p(k)[], h]
        od; p(k)[n] fi
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Mar 24 2020

A[n_, k_] := Module[{h, p}, p[_] = {1}; If[k == 2, If[n == 1, 1, 0], While[ Length[p[k]] < n, For[h = 1 + p[k][[-1]], Mod[k^h, h] != 1, h++]; p[k] = Append[p[k], h]]; p[k][[n]]]];
Table[A[n, 1+d-n], {d, 1, 12}, {n, 1, d}] // Flatten (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

A014959 Integers k such that k divides 22^k - 1.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 39, 49, 63, 81, 117, 147, 189, 243, 273, 343, 351, 441, 507, 567, 729, 819, 1029, 1053, 1143, 1323, 1521, 1701, 1911, 2187, 2401, 2457, 2943, 3081, 3087, 3159, 3429, 3549, 3969, 4401, 4563, 5103, 5733, 6561, 6591, 7203, 7371

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*22^(k-1) (cf. A014940).

Max Alekseyev, Table of n, a(n) for n = 1..69065

Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014960 (b=24).

nxt[{n_,s_}]:={n+1,s+(n+1)*22^n}; Transpose[Select[NestList[nxt,{1,1},7500], Divisible[ Last[#],First[#]]&]][[1]] (* Harvey P. Dale, Jan 27 2015 *)

Edited by Max Alekseyev, Nov 16 2019

A115976 Numbers k that divide 2^(k-2) + 1.

Original entry on oeis.org

1, 3, 49737, 717027, 9723611, 21335267, 32390921, 38999627, 43091897, 86071337, 101848553, 102361457, 228911411, 302948067, 370219467, 393664027, 455781089, 483464027, 1040406177, 1272206987, 2371678553, 2571052241, 2648052857, 3054713937, 3597613307, 3782971499, 3917903851, 4005163577, 5419912241

Offset: 1

Max Alekseyev, Mar 15 2006

nonn

Some larger terms: 4465786944074559659, 1440261542571735083956640176981881665928575750093930787551969

Max Alekseyev, Table of n, a(n) for n = 1..708 (contains all terms below 10^15)

The odd elements of A244673.

Cf. A006521, A006517, A069927, A067945, A067946, A067947, A068382, A068383, A014945, A014946, A014949, A092028.

lst = {}; Do[ If[ PowerMod[2, 2n - 3, 2n - 1] == 2n - 2, AppendTo[lst, 2n - 1]], {n, 10^9}]; lst (* Robert G. Wilson v, Apr 04 2006 *)

More terms from Robert G. Wilson v, Apr 04 2006
Terms a(24) onward from Max Alekseyev, Feb 03 2015
b-file corrected and extended by Max Alekseyev, Oct 27 2018

cp's OEIS Frontend

A127092 Numbers k such that k^2 divides 11^k - 1.

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A177911 Numbers k such that k^3 divides 11^(k^2) - 1.

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A014960 Integers n such that n divides 24^n - 1.

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A014956 Positive integers k such that k divides 14^k - 1.

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A014957 Positive integers k that divide 16^k - 1.

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A333432 A(n,k) is the n-th number m that divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A014959 Integers k such that k divides 22^k - 1.

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A115976 Numbers k that divide 2^(k-2) + 1.

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