A212957 A(n,k) is the number of moduli m such that the multiplicative order of k mod m equals n; square array A(n,k), n>=1, k>=1, read by antidiagonals.

0, 1, 0, 2, 1, 0, 2, 2, 1, 0, 3, 2, 2, 2, 0, 2, 5, 4, 6, 1, 0, 4, 2, 3, 4, 4, 3, 0, 2, 6, 2, 12, 6, 10, 1, 0, 4, 4, 8, 4, 9, 16, 2, 4, 0, 3, 6, 2, 26, 4, 37, 6, 14, 2, 0, 4, 3, 12, 18, 4, 10, 3, 8, 4, 5, 0, 2, 12, 5, 14, 6, 42, 2, 28, 26, 16, 3, 0

Offset: 1

Author

Alois P. Heinz, Jun 01 2012

Keywords

• nonn
• tabl

Examples

			A(4,3) = 6: 3^4 = 81 == 1 (mod m) for m in {5,10,16,20,40,80}.
Square array A(n,k) begins:
  0,  1,  2,  2,  3,  2,  4,  2, ...
  0,  1,  2,  2,  5,  2,  6,  4, ...
  0,  1,  2,  4,  3,  2,  8,  2, ...
  0,  2,  6,  4, 12,  4, 26, 18, ...
  0,  1,  4,  6,  9,  4,  4,  6, ...
  0,  3, 10, 16, 37, 10, 42, 24, ...
  0,  1,  2,  6,  3,  2, 12, 10, ...
  0,  4, 14,  8, 28,  8, 48, 72, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..60
• Wikipedia, Multiplicative order.

Crossrefs

Columns k=1-10 give: A000004, A059499, A059885, A059886, A059887, A059888, A059889, A059890, A059891, A059892.

Rows n=1-10 give: A000005, A059907, A059908, A059909, A059910, A059911, A218256, A218257, A218258, A218259.

Main diagonal gives A252760.

Cf. A000005, A008683.

Programs

Maple

with(numtheory):
A:= (n, k)-> add(mobius(n/d)*tau(k^d-1), d=divisors(n)):
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

Mathematica

a[n_, k_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, k^d - 1], {d, Divisors[n]}]; a[1, 1] = 0; Table[ a[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

PARI

a(n, k) = if(k == 1, 0, sumdiv(n, d, moebius(n/d) * numdiv(k^d-1))); \\ Amiram Eldar, Jan 25 2025

Formula

A(n,k) = |{m : multiplicative order of k mod m = n}|.

A(n,k) = Sum_{d|n} mu(n/d)*tau(k^d-1), mu = A008683, tau = A000005.