A167354 - OEIS

%I #18 Dec 15 2022 06:18:33

%S 1,0,1,0,9,0,25,0,1,0,81,0,121,0,9,0,225,0,289,0,25,0,441,0,81,0,1,0,

%T 729,0,841,0,81,0,225,0,1225,0,121,0,1521,0,1681,0,9,0,2025,0,625,0

%N Totally multiplicative sequence with a(p) = (p-2)^2 = p^2-4p+4 for prime p.

%H G. C. Greubel, <a href="/A167354/b167354.txt">Table of n, a(n) for n = 1..1000</a>

%F Multiplicative with a(p^e) = ((p-2)^2)^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-2)^2)^e(k).

%F a(2k) = 0 for k >= 1.

%F a(n) = A166586(n)^2.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = 1 / (3 * Product_{p prime} (1 + 4/p^2)) = 0.08140990308... . - _Amiram Eldar_, Dec 15 2022

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

%Y Cf. A166586.

%K nonn,mult

%O 1,5

%A _Jaroslav Krizek_, Nov 01 2009