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

Author

Antti Karttunen, Oct 08 2017

Keywords

• nonn

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.

Crossrefs

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

Programs

Mathematica

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

PARI

A293234(n) = sumdiv(n,d,(d
				

Formula

a(n) = Sum_{d|n, dA010052(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