%I #17 Dec 15 2022 06:13:59

%S 1,-5,0,25,16,0,40,-125,0,-80,112,0,160,-200,0,625,280,0,352,400,0,

%T -560,520,0,256,-800,0,1000,832,0,952,-3125,0,-1400,640,0,1360,-1760,

%U 0,-2000,1672,0,1840,2800,0,-2600,2200,0,1600,-1280,0,4000,2800,0,1792

%N Totally multiplicative sequence with a(p) = (p-3)*(p+3) = p^2-9 for prime p.

%H G. C. Greubel, <a href="/A167362/b167362.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = ((p-3)*(p+3))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-3)*(p(k)+3))^e(k).

%F a(n) = A166589(n) * A166591(n).

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 9/p^3 + 9/p^4) = 0.05980933853... . - _Amiram Eldar_, Dec 15 2022

%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 3)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 3)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 11 2016 *)

%Y Cf. A166589, A166591.

%K sign,mult

%O 1,2

%A _Jaroslav Krizek_, Nov 01 2009