A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1

Offset: 1

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

• nonn
• frac
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (16384 terms)

Crossrefs

Cf. A068068, A318314 (denominators).

Differs from A318453 for the first time at n=81, where a(81) = 3, while A318453(81) = 1.

Programs

PARI

up_to = 16384;
A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
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.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318313(k) / A318314(k) ~ 2*n/Pi. - Vaclav Kotesovec, May 10 2025