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

1,16

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

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).

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]

Prog

(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

Crossrefs

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

Sequence in context: A276077 A276935 A235127 * A258059 A093956 A160383

Adjacent sequences: A358342 A358343 A358344 * A358346 A358347 A358348

Keyword

nonn

Author

Amiram Eldar, Nov 11 2022

Status

approved