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

%I A349625 #12 Nov 29 2021 14:29:29
%S A349625 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,
%T A349625 -40,28,4,14,-120,6,14,20,176,27,17,9,8,34,5,41,-48,-88,17,39,50,-46,
%U A349625 -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
%N A349625 Dirichlet convolution of A000027 (identity function) with the Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).
%C A349625 Multiplicative because A000027 and A349623 are.
%H A349625 Antti Karttunen, <a href="/A349625/b349625.txt">Table of n, a(n) for n = 1..20000</a>
%H A349625 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349625 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349625 a(n) = Sum_{d|n} d * A349623(n/d).
%t A349625 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 *)
%o A349625 (PARI)
%o A349625 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349625 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 A349625 A326042(n) = A064989(sigma(A003961(n)));
%o A349625 memoA349623 = Map();
%o A349625 A349623(n) = if(1==n,1,my(v); if(mapisdefined(memoA349623,n,&v), v, v = -sumdiv(n,d,if(d<n,A326042(n/d)*A349623(d),0)); mapput(memoA349623,n,v); (v)));
%o A349625 A349625(n) = sumdiv(n,d,d*A349623(n/d));
%Y A349625 Cf. A000203, A003961, A064989, A326042, A349623, A349624 (Dirichlet inverse).
%K A349625 sign,mult
%O A349625 1,4
%A A349625 _Antti Karttunen_, Nov 26 2021