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

1,16

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

Formula

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

Prog

(PARI)
A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n, n, (A034444(n)-sumdiv(n, d, if((d>1)&&(d<n), A317933perA317934(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);

Crossrefs

Cf. A001790, A034444, A317934 (denominators).

Cf. also A046643, A317831, A317925, A317937.

Sequence in context: A355818 A326538 A333844 * A363329 A370079 A318497

Adjacent sequences: A317930 A317931 A317932 * A317934 A317935 A317936

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 12 2018

Status

approved