A378444 a(n) is the number of divisors d of n such that A083345(d) is even, where A083345(n) is the numerator of Sum(e/p: n=Product(p^e)).

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

Offset: 1

Author

Antti Karttunen, Nov 27 2024

Keywords

• nonn

Comments

Number of terms of A369002 that divide n.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Jon Maiga, Computer-generated formulas for A378444, Sequence Machine.

Crossrefs

Inverse Möbius transform of A369001.

Cf. A000005, A174273, A369002, A378445, A378546.

Cf. also A369257.

Programs

PARI

A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369001(n) = !(A083345(n)%2);
A378444(n) = sumdiv(n,d,A369001(d));

Formula

a(n) = Sum_{d|n} A369001(d).

a(n) = A000005(n) - A378445(n).

a(n) = Sum_{d|n} A023900(d)*A378546(n/d).

a(n) = ceiling(A174273(n)/2). [Conjectured] - Antti Karttunen, May 14 2025