A318662 Denominators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

1, 1, 2, 1, 2, 2, 2, 2, 8, 2, 2, 2, 2, 2, 4, 2, 2, 8, 2, 2, 4, 2, 2, 4, 8, 2, 16, 2, 2, 4, 2, 2, 4, 2, 4, 8, 2, 2, 4, 4, 2, 4, 2, 2, 16, 2, 2, 4, 8, 8, 4, 2, 2, 16, 4, 4, 4, 2, 2, 4, 2, 2, 16, 8, 4, 4, 2, 2, 4, 4, 2, 16, 2, 2, 16, 2, 4, 4, 2, 4, 128, 2, 2, 4, 4, 2, 4, 4, 2, 16, 4, 2, 4, 2, 4, 4, 2, 8, 16, 8, 2, 4, 2, 4, 8

Offset: 1

Author

Antti Karttunen, Sep 03 2018

Keywords

• nonn
• frac

Comments

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A055653, A318661 (numerators), A318663.

Cf. also A046644, A299150, A317932, A317934, A318314.

Programs

PARI

up_to = 1+(2^16);
A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
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&&dA055653(n)));
A318661(n) = numerator(v318661_62[n]);
A318662(n) = denominator(v318661_62[n]);
A318663(n) = valuation(A318662(n),2);

PARI

for(n=1, 100, print1(denominator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 10 2025

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A055653(n) - Sum_{d|n, d>1, d 1.