A308063 Number of ordered factorizations of n into numbers with an odd number of distinct prime divisors.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 5, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 7, 1, 16, 2, 2, 2, 14, 1, 2, 2, 12, 1, 7, 1, 5, 5, 2, 1, 28, 2, 5, 2, 5, 1, 12, 2, 12, 2, 2, 1, 21, 1, 2, 5, 32, 2, 7, 1, 5, 2, 7, 1, 37, 1, 2, 5, 5, 2, 7, 1, 28, 8, 2, 1, 21, 2, 2, 2, 12, 1, 21

Offset: 1

Author

Ilya Gutkovskiy, May 10 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A008578, A030230, A050334, A074206, A092248.

Programs

Mathematica

terms = 90; A[] = 0; Do[A[x] = x + Sum[Boole[OddQ[PrimeNu[k]]] A[x^k], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
f[n_] := f[n] = Boole[OddQ[PrimeNu[n]]]; a[n_] := If[n == 1, n, Sum[If[d < n, f[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 90}]

PARI

a(n) = if(n == 1, 1, sumdiv(n, d, if(dAmiram Eldar, Jul 03 2025

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=1} A(x^A030230(k)).

a(1) = 1; a(n) = Sum_{d|n, dA092248(n/d)*a(d).

a(n) = 1 if n is in A008578.