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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
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Sep 22 2011
STATUS
approved