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

%I A255245 #34 May 22 2025 10:21:42
%S A255245 10,65,140,420,2100,2210,20737,32045,200725,207370,1204350,1347905,
%T A255245 1762645,16502850,31427800,37741340,107671200,130643100,200728169,
%U A255245 239719720,357491225,417225900,430085380,766750575,1088692500,1132409168,1328204850,1788379460
%N A255245 Numbers that divide the average of the squares of their aliquot parts.
%C A255245 Ratio: 1, 1, 5, 10, 78, 1, 109, 565,...
%C A255245 If the ratio is equal to 1 we have 10, 65, 20737 (A140362).
%H A255245 Giovanni Resta, <a href="/A255245/b255245.txt">Table of n, a(n) for n = 1..59</a> (terms < 10^11)
%e A255245 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.
%p A255245 with(numtheory); P:=proc(q) local a,b,k,n;
%p A255245 for n from 2 to q do a:=sort([op(divisors(n))]);
%p A255245 b:=add(a[k]^2,k=1..nops(a)-1)/(nops(a)-1);
%p A255245 if type(b/n,integer) then lprint(n);
%p A255245 fi; od; end: P(10^6);
%t A255245 Select[Range[10^6],Mod[Mean[Most[Divisors[#]^2]],#]==0&] (* _Ivan N. Ianakiev_, Mar 03 2015 *)
%o A255245 (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
%o A255245 (Python)
%o A255245 from __future__ import division
%o A255245 from sympy import factorint
%o A255245 A255245_list = []
%o A255245 for n in range(2,10**9):
%o A255245     s0 = s2 = 1
%o A255245     for p,e in factorint(n).items():
%o A255245         s0 *= e+1
%o A255245         s2 *= (p**(2*(e+1))-1)//(p**2-1)
%o A255245     q, r = divmod(s2-n**2,s0-1)
%o A255245     if not (r or q % n):
%o A255245         A255245_list.append(n) # _Chai Wah Wu_, Mar 08 2015
%Y A255245 Cf. A001065, A255244.
%K A255245 nonn
%O A255245 1,1
%A A255245 _Paolo P. Lava_, Feb 20 2015
%E A255245 More terms from _Michel Marcus_, Feb 20 2015
%E A255245 a(17)-a(24) from _Chai Wah Wu_, Mar 08 2015
%E A255245 a(25)-a(28) from _Giovanni Resta_, May 30 2016