|
%I #17 Dec 15 2022 06:18:50
%S 1,0,5,0,21,0,45,0,25,0,117,0,165,0,105,0,285,0,357,0,225,0,525,0,441,
%T 0,125,0,837,0,957,0,585,0,945,0,1365,0,825,0,1677,0,1845,0,525,0,
%U 2205,0,2025,0,1425,0,2805,0,2457,0,1785,0,3477,0,3717,0,1125
%N Totally multiplicative sequence with a(p) = (p-2)*(p+2) = p^2-4 for prime p.
%H G. C. Greubel, <a href="/A167355/b167355.txt">Table of n, a(n) for n = 1..1000</a>
%F Multiplicative with a(p^e) = ((p-2)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-2)*(p(k)+2))^e(k).
%F a(2k) = 0 for k >= 1.
%F a(n) = A166586(n) * A166590(n).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/Pi^2) / Product_{p prime} (1 - 1/p^2 + 4/p^3 + 4/p^4) = 0.128353657048... . - _Amiram Eldar_, Dec 15 2022
%t a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^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 11 2016 *)
%Y Cf. A166586, A166590.
%K nonn,mult
%O 1,3
%A _Jaroslav Krizek_, Nov 01 2009
|
