A067947 -id:A067947 - cp's OEIS frontend

A127107 Numbers n such that n^2 divides 7^n-1.

1, 2, 3, 4, 6, 8, 12, 20, 24, 40, 57, 60, 100, 114, 120, 156, 200, 220, 228, 258, 300, 312, 440, 456, 516, 600, 660, 780, 1032, 1100, 1140, 1320, 1560, 1640, 1752, 1860, 2172, 2200, 2280, 2580, 2964, 3300, 3660, 3720, 3820, 3900, 4344, 4632, 4902, 4920, 5060

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A127100, A127101, A127102, A127103, A123104, A127105, A127106, A127092. Cf. A067947 = numbers n such that n divides 7^n-1.

Select[Range[10000], IntegerQ[(PowerMod[7, #, #^2 ]-1)/#^2 ]&]

A177907 Numbers k such that k^3 divides 7^(k^2) - 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 20, 24, 30, 40, 50, 57, 60, 68, 78, 100, 110, 111, 114, 120, 136, 150, 156, 200, 204, 220, 222, 228, 258, 300, 312, 330, 340, 390, 408, 440, 444, 456, 516, 550, 570, 600, 660, 680, 780, 820, 876, 888, 930, 1010, 1020, 1032, 1086, 1100

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..2835

Cf. A067947 (k divides 7^k-1), A127107 (k^2 divides 7^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Select[Range[1100],Divisible[7^(#^2)-1,#^3]&] (* Harvey P. Dale, Nov 05 2011 *)

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

A277401 Positive integers n such that 7^n == 2 (mod n).

Original entry on oeis.org

1, 5, 143, 1133, 2171, 8567, 16805, 208091, 1887043, 517295383, 878436591673

Offset: 1

Seiichi Manyama, Oct 13 2016

All terms are odd.

No other terms below 10^15. Some larger terms: 181204957971619289, 21305718571846184078167, 157*(7^157-2)/1355 (132 digits). - Max Alekseyev, Oct 18 2016

			7 == 2 mod 1, so 1 is a term;
16807 == 2 mod 5, so 5 is a term.

Cf. A066438.
Cf. Solutions to 7^n == k (mod n): A277371 (k=-3), A277370 (k=-2), A015954 (k=-1), A067947 (k=1), this sequence (k=2), A277554 (k=3).
Cf. Solutions to b^n == 2 (mod n): A015919 (b=2), A276671 (b=3), A130421 (b=4), A124246 (b=5), this sequence (b=7), A116622 (b=13).

Join[{1},Select[Range[5173*10^5],PowerMod[7,#,#]==2&]] (* The program will generate the first 10 terms of the sequence; it would take a very long time to generate the 11th term. *) (* Harvey P. Dale, Apr 15 2020 *)

isok(n) = Mod(7, n)^n == 2; \\ Michel Marcus, Oct 13 2016

A066438(a(n)) = 2 for n > 1.

a(10) from Michel Marcus, Oct 13 2016

a(11) from Max Alekseyev, Oct 18 2016

A277370 Positive integers k that divide 7^k + 2.

Original entry on oeis.org

1, 3, 15, 69, 2155, 34073, 876047637, 97090036327, 420397381695, 2125899832395, 3177544777277, 34434175473881, 40845965389135, 7267074621260963, 11720938824295035, 21419515204636141

Offset: 1

Seiichi Manyama, Oct 11 2016

All terms are odd.

Some larger terms: 5623143546839445899891, 46186634668308298262543001. - Max Alekseyev, Oct 18 2016

			7^3 + 2 = 345 = 3 * 115, so 3 is a term.

Cf. A066438.

Cf. Solutions to 7^n == k (mod n): A277371 (k=-3), this sequence (k=-2), A015954 (k=-1), A067947 (k=1), A277401 (k=2), A277554 (k=3).

Select[Range[1, 9999, 2], Divisible[7^# + 2, #] &] (* Alonso del Arte, Oct 11 2016 *)

is(n) = Mod(7, n)^n==-2 \\ Felix Fröhlich, Oct 14 2016

A066438(a(n)) = a(n) - 2 for n > 1.

a(8)-a(13) from Max Alekseyev, Oct 18 2016

a(14)-a(16) from Max Alekseyev, Dec 27 2024

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 *)

A277554 Positive integers n such that 7^n == 3 (mod n).

Original entry on oeis.org

1, 2, 46, 2227, 6684830083, 12827743861, 151652531182, 155657642297, 3102126273955, 11006109076099, 50473807426174, 172794904196354

Offset: 1

Max Alekseyev, Oct 19 2016

No other terms below 10^15.

Cf. A066438, A277126.
Cf. Solutions to 7^n == k (mod n): A277371 (k=-3), A277370 (k=-2), A015954 (k=-1), A067947 (k=1), A277401 (k=2).
Cf. Solutions to b^n == 3 (mod n): A050259 (b=2), A130422 (b=4), A123061 (b=5), A116629 (b=13).

isok(n) = Mod(7, n)^n == 3; \\ Michel Marcus, Oct 20 2016

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

cp's OEIS Frontend

A127107 Numbers n such that n^2 divides 7^n-1.

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Mathematica

A177907 Numbers k such that k^3 divides 7^(k^2) - 1.

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Mathematica

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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A277401 Positive integers n such that 7^n == 2 (mod n).

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A277370 Positive integers k that divide 7^k + 2.

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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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