A255244 Numbers that divide the average of the sum of the squares of their divisors.

1, 65, 175, 1105, 5425, 20737, 32045, 70525, 103685, 171275, 200725, 207553, 352529, 372775, 1037765, 1198925, 1264957, 1347905, 1762645, 1824877, 2609425, 2698189, 3628975, 3928475, 4966975, 6324785, 6337175, 8646625, 8813225, 9124385, 10223341, 12774139, 13490945

Offset

1,2

Links

Chai Wah Wu and Giovanni Resta, Table of n, a(n) for n = 1..328 (terms < 10^12, first 82 terms from Chai Wah Wu)

Example

Divisors of 65 are 1, 5, 13, 65. The average of the sum of their squares is (1^2 + 5^2 + 13^2 + 65^2) / 4 = (1 + 25 + 169 + 4225) / 4 = 4420 / 4 = 1105 and 1105 / 65 = 17.

Maple

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

Mathematica

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

Prog

(PARI) isok(n) = (q=sumdiv(n, d, d^2)/numdiv(n)) && (type(q)=="t_INT") && ((q % n) == 0); \\ Michel Marcus, Feb 20 2015
(Python)
from __future__ import division
from sympy import factorint
A255244_list = []
for n in range(1, 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, s0)
....if not (r or q % n):
........A255244_list.append(n) # Chai Wah Wu, Mar 08 2015

Crossrefs

Cf. A000203, A255245.

Sequence in context: A044397 A044778 A054902 * A261989 A051968 A242318

Adjacent sequences: A255241 A255242 A255243 * A255245 A255246 A255247

Keyword

nonn

Author

Paolo P. Lava, Feb 20 2015

Extensions

More terms from Michel Marcus, Feb 20 2015

a(31)-a(33) corrected by Chai Wah Wu, Mar 08 2015

Status

approved