login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A255245
Numbers that divide the average of the squares of their aliquot parts.
2
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
1,1
COMMENTS
Ratio: 1, 1, 5, 10, 78, 1, 109, 565,...
If the ratio is equal to 1 we have 10, 65, 20737 (A140362).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..59 (terms < 10^11)
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A033863 A233246 A229996 * A210369 A058920 A263472
KEYWORD
nonn
AUTHOR
Paolo P. Lava, Feb 20 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
STATUS
approved