A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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

Offset

1,4

Comments

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Links

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

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

a(n) = 2^A318499(n).

Prog

(PARI)
up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)), factor(n)[, 2]));
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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

Crossrefs

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

Sequence in context: A370077 A370080 A372331 * A093997 A157196 A300410

Adjacent sequences: A318495 A318496 A318497 * A318499 A318500 A318501

Keyword

nonn,frac

Author

Antti Karttunen, Aug 30 2018

Status

approved