A333432 -id:A333432 - cp's OEIS frontend

A116621 Positive integers n such that 13^n == 1 (mod n).

1, 2, 3, 4, 6, 8, 9, 12, 14, 16, 18, 20, 24, 27, 28, 32, 36, 40, 42, 48, 54, 56, 60, 64, 68, 72, 80, 81, 84, 96, 98, 100, 108, 112, 120, 126, 128, 136, 140, 144, 160, 162, 168, 180, 183, 192, 196, 200, 204, 216, 220, 224, 240, 243, 252, 256, 272, 280, 288, 294, 300, 320

Offset: 1

Zak Seidov, Feb 19 2006

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)

Column k=13 of A333432.

Cf. A116609, A116611, A116620.

Join[{1}, Select[Range[1, 750], Mod[13^#, #] == 1 &]] (* G. C. Greubel, Nov 19 2017 *)
Join[{1}, Select[Range[320], PowerMod[13, #, #] == 1 &]] (* Robert Price, Apr 10 2020 *)

isok(n) = Mod(13, n)^n == 1; \\ Michel Marcus, Nov 19 2017

1 prepended by Max Alekseyev, Jun 28 2011

A333429 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, 3, 0, 1, 2, 9, 0, 1, 5, 10, 27, 0, 1, 2, 25, 50, 81, 0, 1, 7, 3, 125, 250, 171, 0, 1, 2, 49, 9, 205, 1250, 243, 0, 1, 3, 10, 203, 21, 625, 5050, 513, 0, 1, 2, 9, 50, 343, 26, 1025, 6250, 729, 0, 1, 11, 5, 27, 250, 1379, 27, 2525, 11810, 1539, 0

Offset: 1

Alois P. Heinz, Mar 20 2020

			Square array A(n,k) begins:
  1,    1,     1,    1,   1,    1,     1,   1,    1,     1, ...
  2,    3,     2,    5,   2,    7,     2,   3,    2,    11, ...
  0,    9,    10,   25,   3,   49,    10,   9,    5,   121, ...
  0,   27,    50,  125,   9,  203,    50,  27,   25,   253, ...
  0,   81,   250,  205,  21,  343,   250,  57,   82,  1331, ...
  0,  171,  1250,  625,  26, 1379,  1250,  81,  125,  2783, ...
  0,  243,  5050, 1025,  27, 1421,  2810, 171,  625,  5819, ...
  0,  513,  6250, 2525,  63, 2401,  5050, 243, 2525, 11891, ...
  0,  729, 11810, 3125,  81, 5887,  6250, 513, 3125, 14641, ...
  0, 1539, 25250, 5125, 147, 9653, 14050, 729, 3362, 30613, ...

Alois P. Heinz, Antidiagonals n = 1..20, flattened

Columns k=1-16 give: A130779 (for n>=1), A006521, A015949, A015950, A015951, A015953, A015954, A015955, A015957, A015958, A015960, A015961, A015963, A015965, A015968, A015969.
Rows n=1-2 give: A000012, A092067.
Main diagonal gives A333430.
Cf. A333432.

A:= proc() local h, p; p:= proc() [1] end;
      proc(n, k) if k=1 then `if`(n<3, n, 0) else
        while nops(p(k)) 0 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);

dmax = 12;
mmax = 2^(dmax+3);
col[k_] := col[k] = Select[Range[mmax], Divisible[k^#+1, #]&];
A[n_, k_] := If[n>2 && k==1, 0, col[k][[n]]];
Table[A[n, d-n+1], {d, 1, dmax}, {n, 1, d}] // Flatten (* Jean-François Alcover, Jan 05 2021 *)

A333500 A(n,k) is the n-th number m such that m^2 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, 21, 20, 0, 6, 1, 5, 4, 903, 220, 0, 7, 1, 2, 1555, 6, 2667, 1220, 0, 8, 1, 7, 3, 9673655, 12, 7077, 2420, 0, 9, 1, 2, 889, 4, 187159211791705, 42, 113799, 5060, 0, 10, 1, 3, 4, 2359, 6, 776119592182705, 52, 114681, 13420, 0, 11

Offset: 1

Seiichi Manyama, Mar 24 2020

			Square array A(n,k) begins:
  1, 1,    1,    1,  1,               1, ...
  2, 0,    2,    3,  2,               5, ...
  3, 0,    4,   21,  4,            1555, ...
  4, 0,   20,  903,  6,         9673655, ...
  5, 0,  220, 2667, 12, 187159211791705, ...
  6, 0, 1220, 7077, 42, 776119592182705, ...

Columns k=1-24 give: A000027, A063524, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.
Main diagonal gives A333502.
Cf. A333432.

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", ")))

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A116621 Positive integers n such that 13^n == 1 (mod n).

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A333429 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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A333500 A(n,k) is the n-th number m such that m^2 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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A333433 a(n) is the n-th number m that divides n^m - 1 (or 0 if m does not exist).

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A333506 Numbers k that divide 19^k-1.

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