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A349623 Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).

%I A349623 #13 Nov 29 2021 14:29:00
%S A349623 1,-1,-2,-10,-1,2,-2,18,-25,1,-5,20,-4,2,2,46,-3,25,-2,10,4,5,-6,-36,
%T A349623 -33,4,86,20,-1,-2,-17,-220,10,3,2,250,-10,2,8,-18,-7,-4,-2,50,25,6,
%U A349623 -8,-92,-81,33,6,40,-6,-86,5,-36,4,1,-29,-20,-13,17,50,-886,4,-10,-4,30,12,-2,-31,-450,-3,10,66,20,10,-8,-10
%N A349623 Dirichlet inverse of A326042, where A326042(n) = A064989(sigma(A003961(n))).
%C A349623 Multiplicative because A326042 is.
%H A349623 Antti Karttunen, <a href="/A349623/b349623.txt">Table of n, a(n) for n = 1..20000</a>
%H A349623 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A349623 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A349623 a(1) = 1; a(n) = -Sum_{d|n, d < n} A326042(n/d) * a(d).
%t A349623 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[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349623 (PARI)
%o A349623 A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o A349623 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 A349623 A326042(n) = A064989(sigma(A003961(n)));
%o A349623 memoA349623 = Map();
%o A349623 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)));
%Y A349623 Cf. A000203, A003961, A064989, A326042, A349624, A349625, A349626.
%K A349623 sign,mult
%O A349623 1,3
%A A349623 _Antti Karttunen_, Nov 26 2021