A349437 Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

1, -1, -1, 0, -2, 2, -2, 0, -2, 4, -4, 0, -2, 4, 2, 0, -4, 4, -2, 0, 2, 8, -4, 0, -6, 4, -4, 0, -6, -4, -2, 0, 4, 8, 4, 0, -6, 4, 2, 0, -4, -4, -2, 0, 4, 8, -4, 0, -10, 12, 4, 0, -6, 8, 8, 0, 2, 12, -6, 0, -2, 4, 4, 0, 4, -8, -6, 0, 4, -8, -4, 0, -2, 12, 6, 0, 8, -4, -6, 0, -8, 8, -4, 0, 8, 4, 6, 0, -6, -8, 4, 0, 2

Offset: 1

Author

Antti Karttunen, Nov 18 2021

Keywords

• sign

Comments

Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.

Links

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

Crossrefs

Cf. A055615, A064989, A252463, A349438 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A348045.

Programs

Mathematica

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

PARI

A055615(n) = (n*moebius(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A349437(n) = sumdiv(n,d,A055615(n/d)*A252463(d));

Formula

a(n) = Sum_{d|n} A055615(n/d) * A252463(d).