A358345 a(n) is the number of even square divisors of n.

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, 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, 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

Offset: 1

Author

Amiram Eldar, Nov 11 2022

Keywords

• nonn

Comments

First differs from A235127 at n = 36.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

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]

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))};

PARI

a(n) = sumdiv(n, d, if (issquare(d) && !(d%2), 1)); \\ Michel Marcus, Nov 11 2022

Formula

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

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

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