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

1,4

Links

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

Formula

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

a(1) = 1; a(n) = Sum_{d|n, d<n} A092248(n/d)*a(d).

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

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}]

Prog

(PARI) a(n) = if(n == 1, 1, sumdiv(n, d, if(d<n, (omega(n/d) % 2) * a(d)))); \\ Amiram Eldar, Jul 03 2025

Crossrefs

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

Sequence in context: A317545 A319002 A050363 * A166974 A292504 A281118

Adjacent sequences: A308060 A308061 A308062 * A308064 A308065 A308066

Keyword

nonn

Author

Ilya Gutkovskiy, May 10 2019

Status

approved