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%I #13 Nov 29 2021 14:30:39
%S 1,0,1,10,0,0,1,-8,27,0,4,10,3,0,0,46,2,0,1,0,1,0,5,-8,33,0,-7,10,0,0,
%T 16,6,4,0,0,270,9,0,3,0,6,0,1,40,0,0,7,46,83,0,2,30,5,0,0,-8,1,0,28,0,
%U 12,0,27,1036,0,0,3,20,5,0,30,-216,2,0,33,10,4,0,9,0,447,0,11,10,0,0,0,-32,24,0,3,50,16
%N Möbius transform of A326042, where A326042(n) = A064989(sigma(A003961(n))).
%C Dirichlet convolution of Euler phi (A000010) with A349624.
%C Multiplicative because A326042 is.
%H Antti Karttunen, <a href="/A349626/b349626.txt">Table of n, a(n) for n = 1..20000</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F a(n) = Sum_{d|n} A008683(n/d) * A326042(d).
%F a(n) = Sum_{d|n} A000010(n/d) * A349624(d).
%t 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, s[#] * MoebiusMu[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 27 2021 *)
%o (PARI)
%o A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
%o 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 A326042(n) = A064989(sigma(A003961(n)));
%o A349626(n) = sumdiv(n,d,moebius(n/d)*A326042(d));
%Y Cf. A000010, A003961, A008683, A064989, A326042, A349623, A349624.
%K sign,mult
%O 1,4
%A _Antti Karttunen_, Nov 26 2021
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