A092067 -id:A092067 - cp's OEIS frontend

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.

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 *)

A102457 Least k >= 2 with n^(kn) == n (mod kn), also n^(kn-1) == 1 (mod k).

Original entry on oeis.org

80519, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2, 83, 2, 5, 2, 3, 2, 89, 2, 7, 2, 3

Offset: 2

David W. Wilson, Jan 09 2005

nonn

Motivated by even base-2 pseudoprime 161038, I inquired into base-n pseudoprimes kn that are multiples of n, i.e., n^(kn) == n (mod kn). This is equivalent to n^(kn-1) == 1 (mod k) [W. Edwin Clark] and is satisfied by any k dividing n-1 [Michael Reid]. For n >= 3, this guarantees the existence of a(n) with 2 <= a(n) = k <= lpf(n-1) (lpf = least prime factor). For most n, a(n) = lpf(n-1), exceptional n and a(n) are noted in A102458 and A102459.

Antti Karttunen, Table of n, a(n) for n = 2..12620
Antti Karttunen, Data supplement: n, a(n) computed for n = 2..100000

Cf. A102458, A102459.

Cf. A092067. - R. J. Mathar, Aug 30 2008

Array[Block[{k = 2}, While[PowerMod[#, k # - 1, k] != 1, k++]; k] &, 93, 2] (* Michael De Vlieger, Nov 13 2018 *)

A102457(n) = { for(k=2, oo, if(1==(Mod(n, k)^((k*n)-1)), return(k)); ); } \\ Antti Karttunen, Nov 10 2018

A092028 a(n) is the smallest m > 1 such that m divides n^m-1.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 19, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 37, 2, 3, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2

Offset: 3

Farideh Firoozbakht, Mar 26 2004

nonn

Each prime factor of n-1 is a solution of the equation Mod[n^x-1,x]=0, so a(n) is not greater than smallest prime factor of n-1. Conjecture 1: All terms of this sequence are primes. Conjecture 2: a(n) is the smallest prime factor of n-1 or For n>2 A092028(n)=A020639(n-1).

			a(8)=7 because 7 divides 8^7-1 and there doesn't exist an m such that 1<m<7 and m divides 8^m-1.

Charles R Greathouse IV, Table of n, a(n) for n = 3..10000

Cf. A020639, A092067.

a[n_] := (For[k=2, Mod[n^k-1, k]>0, k++ ];k);Table[a[n], {n, 3, 75}]

a(n)=if(n%2, return(2)); my(m=3); while(Mod(n,m)^m!=1, m+=2); m \\ Charles R Greathouse IV, May 29 2014

a[n_] := (For[k=2, Mod[n^k-1, k]>0, k++ ];k)

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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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A102457 Least k >= 2 with n^(kn) == n (mod kn), also n^(kn-1) == 1 (mod k).

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A092028 a(n) is the smallest m > 1 such that m divides n^m-1.

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