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

%I A349624 #14 Nov 29 2021 14:29:15
%S A349624 1,-1,-1,9,-4,1,-5,-19,23,4,-6,-9,-9,5,4,43,-14,-23,-17,-36,5,6,-17,
%T A349624 19,29,9,-65,-45,-28,-4,-14,-43,6,14,20,207,-27,17,9,76,-34,-5,-41,
%U A349624 -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
%N A349624 Dirichlet convolution of A326042 with A055615 (Dirichlet inverse of n), where A326042(n) = A064989(sigma(A003961(n))).
%C A349624 Multiplicative because A055615 and A326042 are.
%H A349624 Antti Karttunen, <a href="/A349624/b349624.txt">Table of n, a(n) for n = 1..20000</a>
%H A349624 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349624 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349624 a(n) = Sum_{d|n} A055615(d) * A326042(n/d).
%F A349624 For all n >= 1, Sum_{d|n, d<n} = d * a(n/d) = A326042(n) - n = -A348736(n).
%t A349624 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 *)
%o A349624 (PARI)
%o A349624 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349624 A055615(n) = (n*moebius(n));
%o A349624 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 A349624 A326042(n) = A064989(sigma(A003961(n)));
%o A349624 A349624(n) = sumdiv(n,d,A055615(n/d)*A326042(d));
%Y A349624 Cf. A000203, A003961, A055615, A064989, A326042, A349625 (Dirichlet inverse), A349626.
%Y A349624 Cf. also A348736, A349573.
%K A349624 sign,mult
%O A349624 1,4
%A A349624 _Antti Karttunen_, Nov 26 2021