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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.

%I A349437 #10 Nov 21 2021 01:19:15
%S A349437 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,
%T A349437 -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,
%U A349437 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
%N 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.
%C A349437 Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.
%H A349437 Antti Karttunen, <a href="/A349437/b349437.txt">Table of n, a(n) for n = 1..20000</a>
%H A349437 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A349437 a(n) = Sum_{d|n} A055615(n/d) * A252463(d).
%t A349437 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 *)
%o A349437 (PARI)
%o A349437 A055615(n) = (n*moebius(n));
%o A349437 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)};
%o A349437 A252463(n) = if(!(n%2),n/2,A064989(n));
%o A349437 A349437(n) = sumdiv(n,d,A055615(n/d)*A252463(d));
%Y A349437 Cf. A055615, A064989, A252463, A349438 (Dirichlet inverse), A349439 (sum with it).
%Y A349437 Cf.  also A000010, A348045.
%K A349437 sign
%O A349437 1,5
%A A349437 _Antti Karttunen_, Nov 18 2021