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

1,8

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

Formula

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

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 *)

Prog

(PARI) A293235(n) = sumdiv(n, d, (d<n)*ispower(d, 2)*d);

Crossrefs

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

Sequence in context: A347398 A333751 A323921 * A066803 A089608 A250131

Adjacent sequences: A293232 A293233 A293234 * A293236 A293237 A293238

Keyword

nonn

Author

Antti Karttunen, Oct 08 2017

Status

approved