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
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
Programs
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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);
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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