A349624 Dirichlet convolution of A326042 with A055615 (Dirichlet inverse of n), where A326042(n) = A064989(sigma(A003961(n))).

1, -1, -1, 9, -4, 1, -5, -19, 23, 4, -6, -9, -9, 5, 4, 43, -14, -23, -17, -36, 5, 6, -17, 19, 29, 9, -65, -45, -28, -4, -14, -43, 6, 14, 20, 207, -27, 17, 9, 76, -34, -5, -41, -54, -92, 17, -39, -43, 71, -29, 14, -81, -47, 65, 24, 95, 17, 28, -30, 36, -48, 14, -115, 981, 36, -6, -63, -126, 17, -20, -40, -437, -70, 27

Offset: 1

Author

Antti Karttunen, Nov 26 2021

Keywords

• sign
• mult

Comments

Multiplicative because A055615 and A326042 are.

Links

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

Crossrefs

Cf. A000203, A003961, A055615, A064989, A326042, A349625 (Dirichlet inverse), A349626.

Cf. also A348736, A349573.

Programs

Mathematica

f1[p_, e_] := NextPrime[p]^e; s1[1] = 1; s1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[2, e_] := 1; f2[p_, e_] := NextPrime[p, -1]^e; s2[1] = 1; s2[n_] := Times @@ f2 @@@ FactorInteger[n]; s[n_] := s2[DivisorSigma[1, s1[n]]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

PARI

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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)};
A326042(n) = A064989(sigma(A003961(n)));
A349624(n) = sumdiv(n,d,A055615(n/d)*A326042(d));

Formula

a(n) = Sum_{d|n} A055615(d) * A326042(n/d).

For all n >= 1, Sum_{d|n, dA326042(n) - n = -A348736(n).