A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

8, 22, 27, 32, 77, 125, 128, 243, 343, 494, 512, 611, 660, 1073, 1281, 1331, 1425, 2033, 2048, 2187, 2197, 2332, 3125, 4172, 4565, 4913, 5293, 6031, 6859, 8192, 9983, 12167, 13969, 15818, 15947, 16807, 17485, 19683, 23489, 23840, 24389, 25241, 25389, 29791, 32768

Offset: 1

Author

Paolo P. Lava, Mar 22 2018

Keywords

• nonn
• easy

Comments

Semiprimes in the sequence: 22, 77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 98099, 99101, .... - Robert Israel, Mar 29 2018

2^k is a term for all odd k > 1. - Michael S. Branicky, Aug 22 2021

Examples

			Aliquot parts of 77 are 1, 7, 11. Then (1^2 + 7^2 + 11^2)/(1 + 7 + 11) = 171/19 = 9.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1009 (terms below 10^9, terms 1..100 from Paolo P. Lava)

Crossrefs

Cf. A001065, A001157, A020487, A067558.

Contains A056824.

Programs

Maple

with(numtheory): P:=proc(n)
if not isprime(n) and frac((add(p^2,p=divisors(n))-n^2)/(sigma(n)-n))=0
then n; fi; end: seq(P(i),i=2..35*10^3);

Mathematica

aQ[n_] := CompositeQ[n] && Divisible[DivisorSigma[2, n] - n^2, DivisorSigma[1, n] - n]; Select[Range[33000], aQ] (* Amiram Eldar, Aug 17 2019 *)

PARI

isok(n) = (n!=1) && !isprime(n) && (((sigma(n,2) - n^2) % (sigma(n) - n)) == 0); \\ Michel Marcus, Mar 23 2018

Python

from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return len(divs) > 1 and sum(d**2 for d in divs)%sum(divs) == 0
print(list(filter(ok, range(4, 32769)))) # Michael S. Branicky, Aug 22 2021