A378526 Dirichlet inverse of A378548, where A378548 is the sum of divisors d of n such that n/d is odd with an even number of prime factors (counted with multiplicity).

1, -2, -3, 0, -5, 6, -7, 0, -1, 10, -11, 0, -13, 14, 14, 0, -17, 2, -19, 0, 20, 22, -23, 0, -1, 26, 3, 0, -29, -28, -31, 0, 32, 34, 34, 0, -37, 38, 38, 0, -41, -40, -43, 0, 8, 46, -47, 0, -1, 2, 50, 0, -53, -6, 54, 0, 56, 58, -59, 0, -61, 62, 10, 0, 64, -64, -67, 0, 68, -68, -71, 0, -73, 74, 8, 0, 76, -76, -79, 0, 0, 82

Offset: 1

Author

Antti Karttunen, Dec 01 2024

Keywords

• sign

Comments

Agrees with A378525 on all odd n, and also on some even n: 2, 16, 32, 64, 96, 128, 160, 192, ...

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A023900, A055615, A353557, A358777, A369454, A378548.

Cf. also A378525, A378527.

Programs

PARI

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
A378548(n) = sumdiv(n,d,d*A353557(n/d));
memoA378526 = Map();
A378526(n) = if(1==n,1,my(v); if(mapisdefined(memoA378526,n,&v), v, v = -sumdiv(n,d,if(dA378548(n/d)*A378526(d),0)); mapput(memoA378526,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378548(n/d) * a(d).

a(n) = Sum_{d|n} A023900(d)*A369454(n/d).

a(n) = Sum_{d|n} A055615(d)*A358777(n/d).