A378753 Dirichlet inverse of A378752, where A378752(n) = 2*sigma(n) - sigma(A003961(n)), and A003961 is fully multiplicative with a(prime(i)) = prime(i+1).

1, -2, -2, 3, -4, 8, -4, 6, 9, 12, -10, 2, -10, 16, 16, 24, -16, 2, -16, -4, 24, 24, -18, 32, 11, 28, 48, 4, -28, -32, -26, 96, 28, 36, 32, 83, -34, 40, 36, 20, -40, -32, -40, -22, 4, 48, -42, 152, 35, -2, 40, -14, -48, 80, 48, 64, 48, 60, -58, 80, -56, 64, 44, 384, 56, -80, -64, -40, 60, -64, -70, 370, -68, 76, 18

Offset

1,2

Links

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

Index entries for sequences related to prime indices in the factorization of n.

Index entries for sequences related to sigma(n).

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A378752(n/d) * a(d).

Prog

(PARI)
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A378752(n) = ((2*sigma(n)) - sigma(A003961(n)));
memoA378753 = Map();
A378753(n) = if(1==n, 1, my(v); if(mapisdefined(memoA378753, n, &v), v, v = -sumdiv(n, d, if(d<n, A378752(n/d)*A378753(d), 0)); mapput(memoA378753, n, v); (v)));

Crossrefs

Cf. A000203, A003973, A378752 (Dirichlet inverse).

Sequence in context: A302487 A032252 A112708 * A320009 A147558 A032243

Adjacent sequences: A378750 A378751 A378752 * A378754 A378755 A378763

Keyword

sign,new

Author

Antti Karttunen, Dec 11 2024

Status

approved