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A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n). 5
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 (list; graph; refs; listen; history; text; internal format)
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: A103755 A326538 A333844 * A318497 A202150 A093818

Adjacent sequences:  A317930 A317931 A317932 * A317934 A317935 A317936

KEYWORD

nonn,frac,mult

AUTHOR

Antti Karttunen, Aug 12 2018

STATUS

approved

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Last modified July 26 16:44 EDT 2021. Contains 346294 sequences. (Running on oeis4.)