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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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A001065, A255244.

Sequence in context: A033908 A233246 A229996 * A210369 A058920 A263472

Adjacent sequences:  A255242 A255243 A255244 * A255246 A255247 A255248

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

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Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)