A349625 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

1, 1, 1, -8, 4, 1, 5, 2, -22, 4, 6, -8, 9, 5, 4, 50, 14, -22, 17, -32, 5, 6, 17, 2, -13, 9, 20, -40, 28, 4, 14, -120, 6, 14, 20, 176, 27, 17, 9, 8, 34, 5, 41, -48, -88, 17, 39, 50, -46, -13, 14, -72, 47, 20, 24, 10, 17, 28, 30, -32, 48, 14, -110, -1126, 36, 6, 63, -112, 17, 20, 40, -44, 70, 27, -13, -136, 30, 9, 69

Offset: 1

Author

Antti Karttunen, Nov 26 2021

Keywords

• sign
• mult

Comments

Multiplicative because A000027 and A349623 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, A064989, A326042, A349623, A349624 (Dirichlet inverse).

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]]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[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
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)));
memoA349623 = Map();
A349623(n) = if(1==n,1,my(v); if(mapisdefined(memoA349623,n,&v), v, v = -sumdiv(n,d,if(dA326042(n/d)*A349623(d),0)); mapput(memoA349623,n,v); (v)));
A349625(n) = sumdiv(n,d,d*A349623(n/d));

Formula

a(n) = Sum_{d|n} d * A349623(n/d).