A317941 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -11, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Author

Antti Karttunen, Aug 22 2018

Keywords

• sign
• frac
• mult

Comments

Multiplicative because A037445 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A037445, A317934 (denominators).

Cf. also A317933, A317940.

Programs

PARI

up_to = 1+(2^16);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445
v317941aux = DirSqrt(vector(up_to, n, A037445(n)));
A317941(n) = numerator(v317941aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A037445(n) - Sum_{d|n, d>1, d 1.