|
%I #17 Dec 15 2022 06:18:14
%S 1,-3,0,9,12,0,32,-27,0,-36,96,0,140,-96,0,81,252,0,320,108,0,-288,
%T 480,0,144,-420,0,288,780,0,896,-243,0,-756,384,0,1292,-960,0,-324,
%U 1596,0,1760,864,0,-1440,2112,0,1024,-432,0,1260,2700,0,1152,-864,0
%N Totally multiplicative sequence with a(p) = (p+1)*(p-3) = p^2-2p-3 for prime p.
%H G. C. Greubel, <a href="/A167352/b167352.txt">Table of n, a(n) for n = 1..1000</a>
%F 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 a(3k) = 0 for k >= 1.
%F a(n) = A003959(n) * A166589(n).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 + 1/p^2 + 5/p^3 + 3/p^4) = 0.0629795941629... . - _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]] + 1)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* _G. C. Greubel_, Jun 11 2016 *)
%Y Cf. A003959, A166589.
%K sign,mult
%O 1,2
%A _Jaroslav Krizek_, Nov 01 2009
|
