A308063 - OEIS

%I #9 May 11 2019 02:14:31

%S 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,

%T 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,

%U 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

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

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

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

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

%t 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]]

%t 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}]

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

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, May 10 2019