A317935 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003557, n divided by largest squarefree divisor of n.

1, 1, 1, 7, 1, 1, 1, 25, 11, 1, 1, 7, 1, 1, 1, 363, 1, 11, 1, 7, 1, 1, 1, 25, 19, 1, 61, 7, 1, 1, 1, 1335, 1, 1, 1, 77, 1, 1, 1, 25, 1, 1, 1, 7, 11, 1, 1, 363, 27, 19, 1, 7, 1, 61, 1, 25, 1, 1, 1, 7, 1, 1, 11, 9923, 1, 1, 1, 7, 1, 1, 1, 275, 1, 1, 19, 7, 1, 1, 1, 363, 1363, 1, 1, 7, 1, 1, 1, 25, 1, 11, 1, 7, 1, 1, 1, 1335, 1, 27, 11, 133, 1, 1, 1, 25, 1

Offset

1,4

Comments

Multiplicative because A003557 is.

No negative terms among the first 2^20 terms. Is the sequence nonnegative?

Links

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

Wikipedia, Dirichlet convolution

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003557(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]--); factorback(f); }; \\ From A003557
A317935aux(n) = if(1==n, n, (A003557(n)-sumdiv(n, d, if((d>1)&&(d<n), A317935aux(d)*A317935aux(n/d), 0)))/2);
A317935(n) = numerator(A317935aux(n));

Crossrefs

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

Sequence in context: A317936 A277875 A357157 * A360439 A115064 A086868

Adjacent sequences: A317932 A317933 A317934 * A317936 A317937 A317938

Keyword

nonn,frac,mult

Author

Antti Karttunen, Aug 12 2018

Status

approved