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A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.

%I A349356 #19 Nov 27 2021 11:04:06
%S A349356 1,2,2,6,2,4,2,18,8,4,2,12,2,4,4,54,2,16,2,12,4,4,2,36,12,4,32,12,2,8,
%T A349356 2,162,4,4,4,48,2,4,4,36,2,8,2,12,16,4,2,108,16,24,4,12,2,64,4,36,4,4,
%U A349356 2,24,2,4,16,486,4,8,2,12,4,8,2,144,2,4,24,12,4,8,2,108,128,4,2,24,4,4,4,36,2,32,4,12,4
%N A349356 Dirichlet convolution of A003959 with A097945 (Dirichlet inverse of A003958), where A003958 and A003959 are fully multiplicative with a(p) = p-1 and p+1 respectively.
%C A349356 In Dirichlet ring this sequence works as a kind of replacement operator which replaces the factor A003958 with factor A003959. For example, convolving this with A349133 produces A349173.
%H A349356 Antti Karttunen, <a href="/A349356/b349356.txt">Table of n, a(n) for n = 1..20000</a>
%H A349356 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>
%F A349356 a(n) = Sum_{d|n} A003959(n/d) * A097945(d).
%F A349356 Multiplicative with a(p^e) = 2*(p+1)^(e-1). - _Amiram Eldar_, Nov 16 2021
%t A349356 f[p_, e_] := 2*(p + 1)^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 16 2021 *)
%o A349356 (PARI)
%o A349356 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A349356 A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.
%o A349356 A349356(n) = sumdiv(n,d,A003959(n/d)*A097945(d));
%Y A349356 Cf. A003958, A003959, A097945, A349355 (Dirichlet inverse), A349357 (sum with it).
%Y A349356 Cf. also A349133, A349173, A349381.
%K A349356 nonn,mult
%O A349356 1,2
%A A349356 _Antti Karttunen_, Nov 16 2021