A349442 Dirichlet convolution of A000027 (the identity function) with A349350 (Dirichlet inverse of the powerful part of n).

1, 1, 2, -1, 4, 2, 6, -3, -2, 4, 10, -2, 12, 6, 8, -1, 16, -2, 18, -4, 12, 10, 22, -6, -4, 12, -16, -6, 28, 8, 30, 5, 20, 16, 24, 2, 36, 18, 24, -12, 40, 12, 42, -10, -8, 22, 46, -2, -6, -4, 32, -12, 52, -16, 40, -18, 36, 28, 58, -8, 60, 30, -12, 7, 48, 20, 66, -16, 44, 24, 70, 6, 72, 36, -8, -18, 60, 24, 78, -4

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign
• mult

Comments

Multiplicative because both A000027 and A349350 are.

Links

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

Crossrefs

Cf. A000027, A057521, A349350, A349441 (Dirichlet inverse), A349443 (sum with it).

Programs

PARI

A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
memoA349350 = Map();
A349350(n) = if(1==n,1,my(v); if(mapisdefined(memoA349350,n,&v), v, v = -sumdiv(n,d,if(dA057521(n/d)*A349350(d),0)); mapput(memoA349350,n,v); (v)));
A349442(n) = sumdiv(n,d,d*A349350(n/d));

Formula

a(n) = Sum_{d|n} d * A349350(n/d).