A014945 -id:A014945 - cp's OEIS frontend

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

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

A068383 Numbers k such that k divides 11^k - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 16, 18, 20, 24, 25, 30, 32, 36, 40, 42, 48, 50, 54, 60, 64, 72, 80, 84, 90, 96, 100, 108, 114, 120, 125, 126, 128, 144, 150, 156, 160, 162, 168, 180, 192, 200, 210, 216, 222, 228, 240, 244, 250, 252, 256, 270, 272, 288, 294, 300, 312, 320

Offset: 1

Benoit Cloitre, Mar 05 2002

For all k, 2^k, 10^k, 2 * 3^k and 10 * 3^k are in the sequence.

			11^5 - 1 = 161050, which is divisible by 5, so 5 is in the sequence.
11^6 - 1 = 1771560, which is divisible by 6, so 6 is in the sequence.
11^7 = 19487171 = 4 modulo 7, so 7 is not in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014945, A014946, A014949, A014950, A068382.

Join[{1}, Select[Range[500], PowerMod[11, #, #] == 1 &]] (* Robert Price, Apr 04 2020 *)

isok(n) = Mod(11, n)^n == Mod(1, n); \\ Michel Marcus, May 06 2016

def powerMod(a: Int, b: Int, m: Int): Int = b match { case 1 => a % m; case n => a * powerMod(a, n - 1, m) % m }
List(1) ++: (2 to 500).filter(k => powerMod(11, k, k) == 1) // Alonso del Arte, Apr 04 2020

A128356 Least number k > 1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n).

Original entry on oeis.org

20, 21, 1555, 889, 253, 2041, 5846759, 148305659, 1081, 279241, 9641, 950123, 33661, 63213709997, 583223, 3775349, 72707647, 149070763, 196932497, 5091481, 25760459, 14307947980741, 13861, 9362711, 376457, 132766545553, 63757

Offset: 1

Alexander Adamchuk, Mar 02 2007

All listed terms have 2 distinct prime divisors. Most listed terms are semiprimes, except a(7) = 20231*17^2 and a(8) = 410819*19^2. p = prime(n) divides a(n). Quotients a(n)/prime(n) are listed in A128357 = {10, 7, 311, 127, 23, 157, 343927, ...}. a(15) = 583223 = 47*12409. a(16) = 3775349 = 53*71233.

Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.

Cf. A128357 (quotients A128356(n)/prime(n)).

(* This program is not suitable to compute a large number of terms *) a[n_] := For[p = Prime[n]; k = 2, True, k++, If[Length[FactorInteger[k]] == 2, If[Mod[PowerMod[p + 1, k, k] - 1, k] == 0, Print[k]; Return[k]]]]; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Oct 07 2013 *)

Terms a(14) onwards from Max Alekseyev, Feb 08 2010

A128357 Quotients A128356(n)/prime(n).

Original entry on oeis.org

10, 7, 311, 127, 23, 157, 343927, 7805561, 47, 9629, 311, 25679, 821, 1470086279, 12409, 71233, 1232333, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843, 1880415721, 263

Offset: 1

Alexander Adamchuk, Mar 02 2007, Mar 09 2007

A128356 = {20, 21, 1555, 889, 253, 2041, 5846759, ...} = Least number k>1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n). Most listed terms are primes, except a(7) = 20231*17 and a(8) = 410819*19. a(15) = 12409. a(16) = 71233.

Note that all prime listed terms of {a(n)} coincide with terms of A128456 = {2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, ...} = least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n).

Cf. A128356 (least number k > 1 (that is not a power of prime p) such that k divides (p+1)^k-1, where p = prime(n)).
Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.
Cf. A128456 (least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n)).

Terms a(14) onwards from Max Alekseyev, Feb 08 2010

A177805 Numbers k such that k divides 15^k - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 136, 196, 224, 256, 272, 343, 392, 448, 452, 512, 544, 686, 784, 812, 896, 904, 952, 1024, 1088, 1372, 1568, 1624, 1792, 1808, 1904, 2048, 2176, 2312, 2401, 2744, 3136, 3164, 3248, 3584, 3616, 3808, 4096

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

A000420 are the only odd terms of the sequence. - Robert Israel, Feb 25 2020

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000420, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128358, A128360.

Contains A003591.

filter:= n -> 15 &^ n - 1 mod n = 0:
select(filter, [$1..10000]); # Robert Israel, Feb 25 2020

is(n)=Mod(15,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

Original entry on oeis.org

91625794219, 8796093022207, 1557609722332488343, 18216643597893471403

Offset: 1

V. Raman, Oct 04 2012

No other terms below 2^64.
The next term is <= 25790417485109157029391019 = A019320(258). - Emmanuel Vantieghem, Dec 01 2014
Terms A019320(k) belongs to this sequence for k in A297412. - Max Alekseyev, Dec 29 2017

			a(1) = 91625794219 = (2^38 - 2^19 + 1)/3 = A019320(114).
a(2) = 8796093022207 = 2^43 - 1 = A019320(43).

Mersenne Forum, Prime Conjecture

Cf. A069051, A216822, A219037, A217465.

Intersection of A001567 and { 2*A014945(k) + 1 }.

is_A217468(n)={Mod(2,(n-1)*n)^n==2 & !isprime(n)}  \\ - M. F. Hasler, Oct 07 2012

A177807 Numbers k that divide 17^k - 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 42, 48, 54, 60, 64, 72, 78, 80, 84, 96, 100, 108, 116, 120, 126, 128, 144, 156, 160, 162, 168, 180, 192, 200, 216, 220, 232, 234, 240, 252, 256, 288, 294, 300, 312, 320, 324, 336, 342, 348, 360, 378, 384, 400, 420

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

Zizheng Fang, Table of n, a(n) for n = 1..10000
Zizheng Fang, Python program to generate A177807

Cf. A014960, A014956, A014957, A014951, A014949, A014946, A014945, A067945, A128358, A128360, A177805.

{1}~Join~Select[Range[420], PowerMod[17, #, #] == 1 &] (* Giovanni Resta, Jan 30 2020 *)

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 *)

A327840 Numbers m that divide 4^m + 3.

Original entry on oeis.org

1, 7, 16387, 4509253, 24265177, 42673920001, 103949349763, 12939780075073

Offset: 1

Juri-Stepan Gerasimov, Sep 27 2019

Number of solutions < 10^9 to k^n == k-1 (mod n): 1 (if k = 1), 188 (if k = 2, see A006521), 5 (if k = 3, see A015973), 5 (if k = 4, see this sequence), 5 (if k = 5), 10 (if k = 6), 10 (if k = 7), 7 (if k = 8), 5 (if k = 9), 8 (if k = 10), 11 (if k = 11), 8 (if k = 12), 9 (if k = 13), 4 (if k = 14), 3 (if k = 15), 6 (if k = 16), 7 (if k = 17), 7 (if k = 18), ...

a(9) > 10^15. - Max Alekseyev, Nov 10 2022

Solutions to k^n == 1-k (mod n): A006521 (k = 2), A015973 (k = 3), this sequence (k = 4), A123047 (k = 5), A327943 (k = 6).
Solutions to 4^n == k (mod n): A000079 (k = 0), A015950 (k = -1), A014945 (k = 1), A130421 (k = 2), this sequence (k = -3), A130422 (k = 3).
Cf. A015940, A253208.

[1] cat [n: n in [1..10^8] | Modexp(4,n,n) + 3 eq n];

Select[Range[10^7], IntegerQ[(PowerMod[4, #, # ]+3)/# ]&] (* Metin Sariyar, Sep 28 2019 *)

is(n)=Mod(4,n)^n==-3 \\ Charles R Greathouse IV, Sep 29 2019

a(6)-a(7) from Giovanni Resta, Sep 29 2019

a(8) from Max Alekseyev, Nov 10 2022

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

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

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A068383 Numbers k such that k divides 11^k - 1.

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A128356 Least number k > 1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n).

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A128357 Quotients A128356(n)/prime(n).

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A177805 Numbers k such that k divides 15^k - 1.

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A217468 Composite values of n such that 2^n == 2 (mod n*(n-1)).

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A177807 Numbers k that divide 17^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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Mathematica

A327840 Numbers m that divide 4^m + 3.

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