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

Author

Antti Karttunen, Aug 12 2018

Keywords

• nonn
• frac
• mult

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

Crossrefs

Cf. A003557, A046644 (denominators).

Cf. also A300717, A300719, A318317.

Programs

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)&&(dA317935aux(d)*A317935aux(n/d),0)))/2);
A317935(n) = numerator(A317935aux(n));

Formula

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