A358345 - OEIS

%I #13 Nov 12 2022 06:35:33

%S 0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,

%T 0,2,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,3,0,0,0,1,

%U 0,0,0,2,0,0,0,1,0,0,0,2,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,2,0,0,0,2

%N a(n) is the number of even square divisors of n.

%C First differs from A235127 at n = 36.

%C The first position of k >= 0 in this sequence is A187941(k)^2.

%H Amiram Eldar, <a href="/A358345/b358345.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A046951(n) - A298735(n).

%F a(n) = 2 * A046951(n) - A046951(4*n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/24 (A222171).

%t f1[p_, e_] := Floor[e/2] + 1; f2[p_, e_] := If[p == 2, 1, Floor[e/2] + 1]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, if(f[i,1] == 2, 1, 1+f[i,2]\2))};

%o (PARI) a(n) = sumdiv(n, d, if (issquare(d) && !(d%2), 1)); \\ _Michel Marcus_, Nov 11 2022

%Y Cf. A016742, A046951, A187941, A222171, A235127, A298735.

%K nonn

%O 1,16

%A _Amiram Eldar_, Nov 11 2022