A390864 The number of integers k from 1 to n such that gcd(n, k) is a noninfinitary divisor of n.

0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 7, 0, 4, 0, 5, 0, 0, 0, 0, 4, 0, 0, 7, 0, 0, 0, 6, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 0, 21, 6, 8, 0, 13, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 14, 21, 0, 0, 0, 17, 0, 0, 0, 16, 0, 0, 12, 19, 0, 0, 0, 35, 26, 0, 0, 21

Offset

1,9

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = n - A390863(n).

a(n) <= A051953(n), with equality if and only if n = 1.

a(n) = 0 if and only if n is squarefree (A005117).

a(n) = A055654(n) if and only if n is in A138302.

Mathematica

f[p_, e_] := 1 + (p-1) * Total[p^(Select[Range[e], BitAnd[#, e] == # &] - 1)]; a[1] = 0; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); n - prod(i = 1, #f~, 1 + (f[i, 1]-1) * sum(k=1, f[i, 2], if(bitand(k, f[i, 2]) == k, f[i, 1]^(k-1)))); }

Crossrefs

Cf. A005117, A051953, A077609, A138302.

The number of integers k from 1 to n such that gcd(n, k) is a divisor of n of type: A003557 (coreful), A055653 (unitary), A055654 (nonunitary), A010848 (non-coreful), A390863 (infinitary), this sequence (noninfinitary), A390865 (exponential), A390866 (nonexponential), A390867 (bi-unitary), A390868 (non-bi-unitary).

Sequence in context: A385815 A132825 A259480 * A280164 A049597 A210951

Adjacent sequences: A390861 A390862 A390863 * A390865 A390866 A390867

Keyword

nonn,easy

Author

Amiram Eldar, Nov 22 2025

Status

approved