A328856 Number of factorizations of n into distinct numbers with an odd number of distinct prime factors.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 2

Offset: 1

Author

Ilya Gutkovskiy, Oct 28 2019

Keywords

• nonn

Examples

			a(32) = 3 because 32 = 4 * 8 = 2 * 16.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A030230, A045778, A050332, A328855.

Programs

PARI

seq(n)={my(v=vector(n, k, k==1)); for(k=2, n, if(omega(k)%2, my(m=logint(n, k), p=(1 + x + O(x*x^m)), w=vector(n)); for(i=0, m, w[k^i]=polcoef(p, i)); v=dirmul(v, w))); v} \\ Andrew Howroyd, Oct 29 2019, In older versions of PARI, use polcoeff instead of polcoef. - Antti Karttunen, Oct 29 2019

PARI

A328856(n, k=n) = (((n<=k)&&((1==n)||(omega(n)%2))) + sumdiv(n, d, if(d > 1 && d <= k && d < n && (omega(d)%2), A328856(n/d, d-1)))); \\ Antti Karttunen, Oct 29 2019

Formula

Dirichlet g.f.: Product_{k>=1} (1 + A030230(k)^(-s)).

a(n) <= A045778(n). - Antti Karttunen, Oct 29 2019

Extensions

More terms from Antti Karttunen, Oct 29 2019