A391901 The number of distinct factorials that are unitarily dividing n.

1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1

Offset

1,2

Comments

1 = 0! = 1! is counted once.

The only possible terms are 1, 2, and 3.

Links

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

Formula

a(n) <= A055881(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} phi(k!)/k!^2 = Sum_{k>=1} 1/A123476(k) = Sum_{k>=1} A373318(k!)/A373319(k!) = 1.32208875598753557308... .

Mathematica

a[n_] := Module[{f = 1, k = 1, c = 0}, While[f <= n, If[Divisible[n, f] && CoprimeQ[f, n/f], c++]; k++; f *= k]; c]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = 1, k = 1, c = 0); while(f <= n, if(!(n % f) && gcd(f, n/f) == 1, c++); k++; f *= k); c; }

Crossrefs

Cf. A000010 (phi), A000142, A048855, A055881, A077610, A123476, A373318, A373319.

Sequence in context: A175788 A237513 A137844 * A263845 A385731 A079229

Adjacent sequences: A391898 A391899 A391900 * A391902 A391903 A391904

Keyword

nonn,easy

Author

Amiram Eldar, Dec 23 2025

Status

approved