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A195690 Numbers such that the difference between the sum of the even divisors and the sum of the odd divisors is a perfect square. 2

%I

%S 2,6,72,76,162,228,230,238,316,434,530,580,686,690,714,716,756,770,

%T 948,994,1034,1054,1216,1302,1358,1490,1590,1740,1778,1836,1870,1996,

%U 2058,2148,2310,2354,2414,2438,2492,2596,2668,2786,2876,2930,2982,3002,3102

%N Numbers such that the difference between the sum of the even divisors and the sum of the odd divisors is a perfect square.

%C Numbers k such that A002129(k) is a square.

%H Amiram Eldar, <a href="/A195690/b195690.txt">Table of n, a(n) for n = 1..10000</a>

%e The divisors of 76 are { 1, 2, 4, 19, 38, 76}, and (2 + 4 + 38 + 76 ) - (1 + 19 ) = 10^2. Hence 76 is in the sequence.

%p with(numtheory):for n from 2 by 2 to 200 do:x:=divisors(n):n1:=nops(x):s1:=0:s2:=0:for m from 1 to n1 do:if irem(x[m],2)=1 then s1:=s1+x[m]:else s2:=s2+x[m]:fi:od: z:=sqrt(s2-s1):if z=floor(z) then printf(`%d, `,n): else fi:od:

%t f[p_, e_] := If[p == 2, 3 - 2^(e + 1) , (p^(e + 1) - 1)/(p - 1)]; aQ[n_] := IntegerQ[Sqrt[-Times @@ (f @@@ FactorInteger[n])]]; Select[Range[2, 3200], aQ] (* _Amiram Eldar_, Jul 20 2019 *)

%Y Cf. A002129, A195268, A195382.

%K nonn

%O 1,1

%A _Michel Lagneau_, Sep 22 2011

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Last modified December 15 14:37 EST 2019. Contains 329999 sequences. (Running on oeis4.)