A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 9, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 15, 48, 54, 48, 25, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 36, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 24, 84, 144, 68, 72, 96, 144, 72, 33, 74, 114, 64, 80, 96, 168, 80, 54, 34

Offset: 1

Author

Antti Karttunen, Nov 26 2024

Keywords

• sign

Comments

The first negative term is a(2592) = -48.

Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Links

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

Crossrefs

Dirichlet inverse of A378434.

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.

Programs

PARI

A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);
memoA378435 = Map();
A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(dA378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378434(n/d) * a(d).