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, 6, 72, 76, 162, 228, 230, 238, 316, 434, 530, 580, 686, 690, 714, 716, 756, 770, 948, 994, 1034, 1054, 1216, 1302, 1358, 1490, 1590, 1740, 1778, 1836, 1870, 1996, 2058, 2148, 2310, 2354, 2414, 2438, 2492, 2596, 2668, 2786, 2876, 2930, 2982, 3002, 3102

Offset

1,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

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.

Maple

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:

Mathematica

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 *)

Crossrefs

Cf. A002129, A195268, A195382.

Sequence in context: A103527 A239543 A180982 * A359365 A329965 A171582

Adjacent sequences: A195687 A195688 A195689 * A195691 A195692 A195693

Keyword

nonn

Author

Michel Lagneau, Sep 22 2011

Status

approved