A349572 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A048673.

1, 0, 0, -1, 1, -2, 1, -4, -4, -3, 4, -4, 4, -5, -6, -12, 7, -6, 7, -7, -10, -6, 8, -6, -4, -8, -24, -11, 13, -2, 12, -32, -12, -9, -14, -4, 16, -11, -16, -13, 19, -2, 19, -16, -22, -14, 20, -4, -18, -15, -18, -20, 23, -10, -14, -19, -22, -15, 28, 14, 27, -18, -34, -80, -20, -8, 31, -25, -28, -8, 34, 14, 33, -20

Offset: 1

Author

Antti Karttunen, Nov 23 2021

Keywords

• sign

Comments

Also Dirichlet convolution of A349384 with A349388.

Links

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

Crossrefs

Cf. A000027, A048673, A323893, A349384, A349388, A349571 (Dirichlet inverse).

Cf. also A349397.

Programs

Mathematica

f[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2; 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 23 2021 *)

PARI

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
memoA323893 = Map();
A323893(n) = if(1==n,1,my(v); if(mapisdefined(memoA323893,n,&v), v, v = -sumdiv(n,d,if(dA048673(n/d)*A323893(d),0)); mapput(memoA323893,n,v); (v)));
A349572(n) = sumdiv(n,d,d*A323893(n/d));

Formula

a(n) = Sum_{d|n} d * A323893(n/d).

a(n) = Sum_{d|n} A349384(d) * A349388(n/d).