cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Crossrefs

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A128360.
Cf. A177914, A128394.

Programs

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

Extensions

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A014956, A177805, A177807, A128358, A128360
Cf. A128396, A177916

Programs

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

Extensions

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

Views

Author

Seiichi Manyama, Mar 21 2020

Keywords

Examples

			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, ...

Links

Crossrefs

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.

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Olivier Gérard

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Extensions

Edited by Max Alekseyev, Nov 16 2019
Showing 1-5 of 5 results.

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