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%I #17 Dec 15 2022 06:17:57
%S 1,0,4,0,18,0,40,0,16,0,108,0,154,0,72,0,270,0,340,0,160,0,504,0,324,
%T 0,64,0,810,0,928,0,432,0,720,0,1330,0,616,0,1638,0,1804,0,288,0,2160,
%U 0,1600,0
%N Totally multiplicative sequence with a(p) = (p+1)*(p-2) = p^2-p-2 for prime p.
%H G. C. Greubel, <a href="/A167350/b167350.txt">Table of n, a(n) for n = 1..1000</a>
%F Multiplicative with a(p^e) = ((p+1)*(p-2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)+1)*(p(k)-2))^e(k).
%F a(2k) = 0 for k >= 1.
%F a(n) = A003959(n) * A166586(n).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 3/p^3 + 2/p^4) = 0.1140434597... . - _Amiram Eldar_, Dec 15 2022
%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 10 2016 *)
%Y Cf. A003959, A166586.
%K nonn,mult
%O 1,3
%A _Jaroslav Krizek_, Nov 01 2009