A255245 Numbers that divide the average of the squares of their aliquot parts.

10, 65, 140, 420, 2100, 2210, 20737, 32045, 200725, 207370, 1204350, 1347905, 1762645, 16502850, 31427800, 37741340, 107671200, 130643100, 200728169, 239719720, 357491225, 417225900, 430085380, 766750575, 1088692500, 1132409168, 1328204850, 1788379460

Offset: 1

Author

Paolo P. Lava, Feb 20 2015

Keywords

• nonn

Comments

Ratio: 1, 1, 5, 10, 78, 1, 109, 565,...

If the ratio is equal to 1 we have 10, 65, 20737 (A140362).

Examples

			Aliquot parts of 10 are 1, 2, 5. The average of their squares is (1^2 + 2^2 + 5^2) / 3 = (1 + 4 + 25) / 3 = 30 / 3 = 10 and 10 / 10 = 1.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..59 (terms < 10^11)

Crossrefs

Cf. A001065, A255244.

Programs

Maple

with(numtheory); P:=proc(q) local a,b,k,n;
for n from 2 to q do a:=sort([op(divisors(n))]);
b:=add(a[k]^2,k=1..nops(a)-1)/(nops(a)-1);
if type(b/n,integer) then lprint(n);
fi; od; end: P(10^6);

Mathematica

Select[Range[10^6],Mod[Mean[Most[Divisors[#]^2]],#]==0&] (* Ivan N. Ianakiev, Mar 03 2015 *)

PARI

isok(n) = (q=(sumdiv(n, d, (d!=n)*d^2)/(numdiv(n)-1))) && (type(q)=="t_INT") && ((q % n) == 0); \\ Michel Marcus, Feb 20 2015

Python

from _future_ import division
from sympy import factorint
A255245_list = []
for n in range(2,10**9):
    s0 = s2 = 1
    for p,e in factorint(n).items():
        s0 *= e+1
        s2 *= (p**(2*(e+1))-1)//(p**2-1)
    q, r = divmod(s2-n**2,s0-1)
    if not (r or q % n):
        A255245_list.append(n) # Chai Wah Wu, Mar 08 2015

Extensions

More terms from Michel Marcus, Feb 20 2015

a(17)-a(24) from Chai Wah Wu, Mar 08 2015

a(25)-a(28) from Giovanni Resta, May 30 2016