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A359530 Multiplicative with a(p^e) = (p + 4)^e.

%I A359530 #27 Feb 27 2023 01:34:08
%S A359530 1,6,7,36,9,42,11,216,49,54,15,252,17,66,63,1296,21,294,23,324,77,90,
%T A359530 27,1512,81,102,343,396,33,378,35,7776,105,126,99,1764,41,138,119,
%U A359530 1944,45,462,47,540,441,162,51,9072,121,486,147,612,57,2058,135,2376,161
%N A359530 Multiplicative with a(p^e) = (p + 4)^e.
%H A359530 Vaclav Kotesovec, <a href="/A359530/b359530.txt">Table of n, a(n) for n = 1..10000</a>
%F A359530 Dirichlet g.f.: Product_{primes p} 1 / (1 - p^(1-s) - 4*p^(-s)).
%F A359530 Dirichlet g.f.: zeta(s-1) * (1 + 4/(2^s - 6)) * Product_{primes p, p>2} (1 + 4/(p^s - p - 4)).
%F A359530 Sum_{k=1..n} a(k) has an average value 2*c*zeta(r-1) * n^r / (3*log(6)), where r = 1 + log(3)/log(2) = 2.5849625007211561814537389439478165... and c = Product_{primes p, p>2} (1 + 4/(p^r - p - 4)) = 1.5747380964592139...
%t A359530 g[p_, e_] := (p + 4)^e; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]
%o A359530 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-p*X-4*X))[n], ", "))
%o A359530 (Python)
%o A359530 from math import prod
%o A359530 from sympy import factorint
%o A359530 def A359530(n): return prod((p+4)**e for p, e in factorint(n).items()) # _Chai Wah Wu_, Feb 26 2023
%Y A359530 Cf. A166589 (multiplicative with a(p^e) = (p-3)^e), A166586 (p-2), A003958 (p-1), A000027 (p), A003959 (p+1), A166590 (p+2), A166591 (p+3).
%K A359530 nonn,mult
%O A359530 1,2
%A A359530 _Vaclav Kotesovec_, Feb 26 2023