A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 16, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -30, 48, 54, 48, 48, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -64, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 56, 84, -144, -68, -90, 96, -144, -72, -90

Offset: 1

Author

Antti Karttunen, Nov 26 2024

Keywords

• sign

Comments

Arithmetic mean between A158523 and A178450.

Apparently differs from A378433 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

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433, A378435 (Dirichlet inverse).

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);

Formula

a(n) = (1/2) * (A158523(n)+A178450(n)).