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%I #21 Feb 10 2017 01:19:49
%S 2,6,162,230,238,434,530,686,690,714,770,994,1034,1054,1302,1358,1490,
%T 1590,1778,1870,2058,2310,2354,2414,2438,2786,2930,2982,3002,3102,
%U 3162,3290,3298,3374,3410,3542,3830,4074,4202,4318,4402,4470,4718,4806,5334,5510,5610,5798,5990,6014,6286
%N Even numbers k such that half the sum of the even divisors equals the sum of the odd divisors and both are (the same) square.
%C a(n) == 2 mod 4.
%C The corresponding squares are 1, 2^2, 11^2, 12^2, 12^2, 16^2, 18^2, 20^2, 24^2, 24^2, 24^2, 24^2, 24^2, 24^2, 32^2, 28^2, 30^2, 36^2, 32^2, 36^2, 40^2,...
%C There exists a subsequence {a(n)} intersection {A281707} = 6, 434, 1302, 1778, 7874, 23622, 114674, ... of numbers of the form 2p1*p2*...pk where p1, p2,...,pk are Mersenne primes = 3, 7, 31, 127, 8191,... (see A000668).
%C The corresponding squares are also powers of 2: 2^2, 2^8, 2^10, 2^10, 2^12,...
%H Charles R Greathouse IV, <a href="/A281819/b281819.txt">Table of n, a(n) for n = 1..10000</a>
%e 162 is in the sequence because the divisors are {1, 2, 3, 6, 9, 18, 27, 54, 81, 162} => half sum of even divisors = (2 + 6 + 18 + 54 + 162)/2 = 11^2 and sum of odd divisors = 1 + 3 + 9 + 27 + 81 = 11^2.
%p with(numtheory):
%p for n from 2 by 2 to 10^5 do:
%p x:=divisors(n):n1:=nops(x):s0:=0:s1:=0:
%p for k from 1 to n1 do:
%p if irem(x[k],2)=0
%p then
%p s0:=s0+ x[k]:
%p else
%p s1:=s1+ x[k]:
%p fi:
%p od:
%p s11:=sqrt(s1):s22:=sqrt(s0/2):
%p if floor(s11)=s11 and floor(s22)=s22 and s11=s22
%p then
%p printf(`%d, `,n):
%p else
%p fi:
%p od:
%o (PARI) forstep(k=1,1e3,2,if(issquare(sigma(k)), print1(2*k", "))) \\ _Charles R Greathouse IV_, Feb 06 2017
%o (PARI) is(n)=n%4==2 && issquare(sigma(n/2)) \\ _Charles R Greathouse IV_, Feb 06 2017
%Y Cf. A000668, A281707, A006532.
%K nonn
%O 1,1
%A _Michel Lagneau_, Feb 03 2017
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