A317927 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A005187.

1, 3, 2, 19, 4, 2, 11, 63, 6, 3, 19, 13, 23, 17, 5, 867, 16, 4, 35, 5, 17, 25, 21, 11, 31, 29, 13, 113, 27, 13, 57, 3069, 13, 9, 23, 25, 71, 41, 14, 69, 79, 33, 41, 169, 9, 25, 89, 615, 259, 53, 17, 197, 51, 25, 29, 389, 20, 31, 113, 59, 117, 67, 10, 22199, 18, 14, 131, 31, 51, 71, 69, 11, 143, 77, 22, 281, 91, 35, 153, 489, 71, 85, 81, 151, 19

Offset: 1

Author

Antti Karttunen, Aug 11 2018

Keywords

• sign
• frac

Comments

The first negative term is a(330) = -21.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A005187, A317928 (denominators).

Cf. also A297111, A300244, A299151, A317931.

Programs

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A317927perA317928(n) = if(1==n,n,(A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2);
A317927(n) = numerator(A317927perA317928(n));

PARI

\\ Memoized implementation:
memo = Map();
A317927perA317928(n) = if(1==n,n,if(mapisdefined(memo,n),mapget(memo,n),my(v = (A005187(n)-sumdiv(n,d,if((d>1)&&(dA317927perA317928(d)*A317927perA317928(n/d),0)))/2); mapput(memo,n,v); (v)));

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A005187(n) - Sum_{d|n, d>1, d 1.