A293234 a(n) is the number of proper divisors of n that are square.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 2, 3, 1, 1, 1, 2, 1

Offset

1,8

Comments

First occurrence of k: 2, 8, 32, 72, 144, 288, 576, 1152, 2304, 4608, 3600, 7200, 36864, 20736, 14400, 28800, 32400, 64800, 57600, 115200, 663552, 18874368, 129600, 259200, 3359232, 810000, 921600, 1843200, 518400, 1036800, 705600, 1411200, etc. - Robert G. Wilson v, Oct 08 2017

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = Sum_{d|n, d<n} A010052(d).

a(n) = A046951(n) - A010052(n).

G.f.: Sum_{k>=1} x^(2*k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Apr 13 2021

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/6 (A013661). - Amiram Eldar, Dec 01 2023

Mathematica

f[n_] := Length@ Select[ Sqrt@ Most@ Divisors@ n, IntegerQ]; Array[f, 105] (* Robert G. Wilson v, Oct 08 2017 *)

Prog

(PARI) A293234(n) = sumdiv(n, d, (d<n)*ispower(d, 2));

Crossrefs

Cf. A010052, A013661, A046951, A293227, A293235.

Sequence in context: A205119 A355150 A333915 * A347441 A360617 A102097

Adjacent sequences: A293231 A293232 A293233 * A293235 A293236 A293237

Keyword

nonn

Author

Antti Karttunen, Oct 08 2017

Status

approved