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
Keywords
Examples
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.
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)
Programs
-
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 *)
-
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
Extensions
More terms from Michel Marcus, Feb 20 2015
a(31)-a(33) corrected by Chai Wah Wu, Mar 08 2015