A317846 Numerators of rational valued sequence whose Dirichlet convolution with itself yields sequence A051953 (cototient of n) + A063524 (1, 0, 0, 0, ...).

1, 1, 1, 7, 1, 7, 1, 25, 11, 11, 1, 43, 1, 15, 13, 363, 1, 71, 1, 67, 17, 23, 1, 139, 19, 27, 61, 91, 1, 57, 1, 1335, 25, 35, 21, 365, 1, 39, 29, 215, 1, 81, 1, 139, 131, 47, 1, 1875, 27, 199, 37, 163, 1, 367, 29, 291, 41, 59, 1, 235, 1, 63, 171, 9923, 33, 129, 1, 211, 49, 137, 1, 1055, 1, 75, 235, 235, 33, 153, 1, 2883, 1363, 83, 1, 335, 41

Offset: 1

Author

Antti Karttunen, Aug 12 2018

Keywords

• sign
• frac

Comments

The first negative term is a(420) = -1269.

Links

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

Crossrefs

Cf. A051953, A063524, A046644 (denominators).

Cf. also A317845, A317925.

Programs

PARI

A317846aux(n) = if(1==n,n,((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2);
A317846(n) = numerator(A317846aux(n));

PARI

\\ Memoized implementation:
memo317846 = Map();
A317846aux(n) = if(1==n,n,if(mapisdefined(memo317846,n),mapget(memo317846,n),my(v = ((n-eulerphi(n))-sumdiv(n,d,if((d>1)&&(dA317846aux(d)*A317846aux(n/d),0)))/2); mapput(memo317846,n,v); (v)));

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A051953(n) - Sum_{d|n, d>1, d 1.