A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3

Offset

1,64

Links

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

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
up_to = (2^16)+1;
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};
A049599(n) = factorback(apply(e -> (1+numdiv(e)), factor(n)[, 2]));
v318671_72 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_72[n]);

Crossrefs

Cf. A049599, A318672 (denominators).

Sequence in context: A359215 A260237 A118135 * A337335 A109014 A268357

Adjacent sequences: A318668 A318669 A318670 * A318672 A318673 A318674

Keyword

sign,frac

Author

Antti Karttunen, Sep 03 2018

Status

approved