A293235 a(n) is the sum of proper divisors of n that are square.

0, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 14, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 1, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 21, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 21, 1, 50, 10, 30, 1, 1, 1, 5, 1

Offset: 1

Author

Antti Karttunen, Oct 08 2017

Keywords

• nonn

Comments

a(n) = 1 if and only if n > 1 is squarefree or the square of a prime. - Robert Israel, Oct 08 2017

Links

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

Crossrefs

Cf. A010052, A035316, A078434, A293228, A293234.

Programs

Maple

A035316:= n -> mul((p[1]^(p[2]+2-(p[2] mod 2))-1)/(p[1]^2-1), p = ifactors(n)[2]):
f:= n -> A035316(n) - `if`(issqr(n),n,0):
map(f, [$1..100]); # Robert Israel, Oct 08 2017

Mathematica

Table[Total[Select[Most[Divisors[n]],IntegerQ[Sqrt[#]]&]],{n,120}] (* Harvey P. Dale, Dec 29 2023 *)

PARI

A293235(n) = sumdiv(n,d,(d
				

Formula

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

a(n) = A035316(n) - (A010052(n)*n).

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

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3/2)-1)/3 = 0.537458449561... . - Amiram Eldar, Dec 01 2023