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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.
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%I #19 Nov 27 2021 11:04:06

%S 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 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 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 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 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 Antti Karttunen, <a href="/A349356/b349356.txt">Table of n, a(n) for n = 1..20000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>

%F a(n) = Sum_{d|n} A003959(n/d) * A097945(d).

%F Multiplicative with a(p^e) = 2*(p+1)^(e-1). - _Amiram Eldar_, Nov 16 2021

%t 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 (PARI)

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A097945(n) = (moebius(n)*eulerphi(n)); \\ Also Dirichlet inverse of A003958.

%o A349356(n) = sumdiv(n,d,A003959(n/d)*A097945(d));

%Y Cf. A003958, A003959, A097945, A349355 (Dirichlet inverse), A349357 (sum with it).

%Y Cf. also A349133, A349173, A349381.

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Nov 16 2021