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A109624 Totally multiplicative sequence with a(p) = (p-1)*(p+3) = p^2+2p-3 for prime p.

%I A109624 #19 Nov 05 2022 08:17:52
%S A109624 1,5,12,25,32,60,60,125,144,160,140,300,192,300,384,625,320,720,396,
%T A109624 800,720,700,572,1500,1024,960,1728,1500,896,1920,1020,3125,1680,1600,
%U A109624 1920,3600,1440,1980,2304,4000,1760,3600,1932,3500,4608,2860,2300,7500,3600
%N A109624 Totally multiplicative sequence with a(p) = (p-1)*(p+3) = p^2+2p-3 for prime p.
%H A109624 Vaclav Kotesovec, <a href="/A109624/b109624.txt">Table of n, a(n) for n = 1..10000</a>
%F A109624 Multiplicative with a(p^e) = ((p-1)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+3))^e(k).
%F A109624 a(n) = A003958(n) * A166591(n).
%F A109624 Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + 2*p - 4)) = 1.471999388763656342016756485604184156984049961181587531678650682804811302... - _Vaclav Kotesovec_, Sep 20 2020
%F A109624 Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 3/p^2 + 1/p^3 + 3/p^4)) = 0.6324191395... . - _Amiram Eldar_, Nov 05 2022
%t A109624 f[p_, e_] := ((p - 1)*(p + 3))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2022 *)
%o A109624 (PARI) a(n) = {f = factor(n); return (prod(k=1, #f~, ((f[k, 1]-1)*(f[k, 1]+3))^f[k, 2]));} \\ _Michel Marcus_, Jun 13 2013
%Y A109624 Cf. A003958, A166591.
%K A109624 nonn,mult
%O A109624 1,2
%A A109624 _Jaroslav Krizek_, Nov 01 2009