A378228 Dirichlet inverse of A341528, where A341528(n) = n * sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

1, -8, -18, 12, -40, 144, -84, 0, 45, 320, -154, -216, -234, 672, 720, 0, -340, -360, -456, -480, 1512, 1232, -690, 0, 175, 1872, 0, -1008, -928, -5760, -1178, 0, 2772, 2720, 3360, 540, -1554, 3648, 4212, 0, -1804, -12096, -2064, -1848, -1800, 5520, -2538, 0, 539, -1400, 6120, -2808, -3180, 0, 6160, 0, 8208, 7424

Offset: 1

Author

Antti Karttunen, Nov 23 2024

Keywords

• sign
• mult

Comments

Multiplicative because A341528 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization.
• Index entries for sequences related to sigma(n).

Crossrefs

Dirichlet inverse of A341528.

Cf. A003973, A378221.

Cf. also A378229.

Programs

PARI

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
memoA378228 = Map();
A378228(n) = if(1==n,1,my(v); if(mapisdefined(memoA378228,n,&v), v, v = -sumdiv(n,d,if(dA341528(n/d)*A378228(d),0)); mapput(memoA378228,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA341528(n/d) * a(d).