A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Aug 12 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A034444 is.

The first 2^20 terms are positive. Is the sequence nonnegative?

Records seem to be A001790, occurring at A000302 (apart from 4).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Crossrefs

Cf. A001790, A034444, A317934 (denominators).

Cf. also A046643, A317831, A317925, A317937.

Programs

PARI

A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n,n,(A034444(n)-sumdiv(n,d,if((d>1)&&(dA317933perA317934(d)*A317933perA317934(n/d),0)))/2);
A317933(n) = numerator(A317933perA317934(n));

PARI

up_to = 65537;
\\ Faster:
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.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);

PARI

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A034444(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317933(k) / A317934(k) ~ sqrt(6)*n/Pi. - Vaclav Kotesovec, May 10 2025