A128358 -id:A128358 - cp's OEIS frontend

A177807 Numbers k that divide 17^k - 1.

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 *)

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

A333433 a(n) is the n-th number m that divides n^m - 1 (or 0 if m does not exist).

Original entry on oeis.org

1, 0, 4, 21, 8, 1555, 9, 6223, 40, 999, 20, 130801, 24, 4484077, 128, 117, 60, 118285781329, 42, 1432001198261, 104, 819, 72, 302508121, 81, 75625, 200, 61731, 78, 14507145975869, 72, 21958351241, 820, 12321, 289, 4375, 144

Offset: 1

Seiichi Manyama, Mar 21 2020

From Jinyuan Wang, Mar 25 2020: (Start)
For n > 2, n < a(n) < q^(n-1), where q is a prime factor of n - 1.
If p is a prime, then a(p^e+1) is divisible by p. Proof: we can prove that p | m for m > 1 and n = p^e + 1. If n^m == 1 (mod m) and m > 1 is the minimum value that cannot be divisible by p, then gcd(m, eulerphi(m)) = 1. Thus, m must be of the form q*p_2*...*p_k, where q < p_2 < ... < p_k. Note that q | (n^m - 1) = (n^q - 1)*(Sum_{i=0..(m/q)-1} n)^(i*q)) and n^q - 1 can never be divisible by q. Therefore, Sum_{i=0..(m/q)-1} n^(i*q) == n^(m/q) - 1 == 0 (mod q). Because n^(q-1) == 1 (mod q) and gcd(m/q, q - 1) = 1, then n == 1 (mod q), a contradiction! (End)
a(38) <= 14948925257859919. - Giovanni Resta, Apr 15 2020

OEIS Wiki, 2^n mod n

Main diagonal of A333432.

Cf. A014960, A128358, A128360, A333430.

{a(n) = if(n==2, 0, my(cnt=0, k=0); while(cnt

a(n) = A333432(n,n).

a(30)-a(37) from Giovanni Resta, Apr 15 2020

A333506 Numbers k that divide 19^k-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 27, 30, 32, 36, 40, 42, 48, 50, 54, 60, 64, 72, 80, 81, 84, 90, 96, 100, 108, 110, 120, 126, 128, 136, 144, 150, 156, 160, 162, 168, 180, 192, 200, 210, 216, 220, 240, 243, 250, 252, 256, 270, 272, 288, 294, 300, 312, 320, 324, 330, 336

Offset: 1

Seiichi Manyama, Mar 25 2020

nonn

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

Columns k=19 of A333432.

Cf. A128358, A128360, A128399.

Join[{1}, Select[Range[2, 336], PowerMod[19, #, #] == 1 &]] (* Amiram Eldar, May 05 2021 *)

for(k=1, 1e3, if(Mod(19, k)^k==1, print1(k", ")))

A014962 Odd numbers k that divide 25^k - 1.

Original entry on oeis.org

1, 3, 9, 21, 27, 63, 81, 93, 147, 171, 189, 243, 279, 441, 513, 567, 609, 651, 729, 837, 903, 1029, 1197, 1323, 1539, 1701, 1827, 1953, 2187, 2511, 2667, 2709, 2883, 2943, 3087, 3249, 3591, 3969, 4263, 4401, 4557, 4617, 5103, 5301, 5481, 5859, 6321

Offset: 1

Olivier Gérard

nonn

Also, numbers k such that k divides s(k), where s(1)=1, s(j) = s(j-1) + j*25^(j-1).

Equivalently, numbers k that divide ((24*k - 1)*25^k + 1) / 24^2 (cf. A014943).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014943, A014945, A014946, A014949, A014950, A014951, A014956, A014957, A128358, A128360, A014959, A014960.

select(t -> 25 &^ t - 1 mod t = 0, [seq(i,i=1..10^4,2)]); # Robert Israel, Oct 04 2020

Edited by Max Alekseyev, Nov 16 2019

cp's OEIS Frontend

A177807 Numbers k that divide 17^k - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A333433 a(n) is the n-th number m that divides n^m - 1 (or 0 if m does not exist).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A333506 Numbers k that divide 19^k-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A014962 Odd numbers k that divide 25^k - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Extensions