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A088827 Even numbers with odd abundance: even squares or two times squares.

%I A088827 #35 Jul 09 2023 03:15:46
%S A088827 2,4,8,16,18,32,36,50,64,72,98,100,128,144,162,196,200,242,256,288,
%T A088827 324,338,392,400,450,484,512,576,578,648,676,722,784,800,882,900,968,
%U A088827 1024,1058,1152,1156,1250,1296,1352,1444,1458,1568,1600,1682,1764,1800,1922
%N A088827 Even numbers with odd abundance: even squares or two times squares.
%C A088827 Sigma(k)-2k is odd means that sigma(k) is also odd.
%C A088827 Odd numbers with odd abundance are in A016754. Odd numbers with even abundance are in A088828. Even numbers with even abundance are in A088829.
%H A088827 Michael De Vlieger, <a href="/A088827/b088827.txt">Table of n, a(n) for n = 1..10000</a>
%F A088827 Conjecture: a(n) = ((2*r) + 1)^2 * 2^(c+1) where r and c are the corresponding row and column of n in the table format of A191432, where the first row and column are 0. - _John Tyler Rascoe_, Jul 12 2022
%F A088827 Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003). - _Amiram Eldar_, Jul 09 2023
%e A088827 From _Michael De Vlieger_, May 14 2017: (Start)
%e A088827 4 is a term since it is even and the sum of its divisors {1,2,4} = 7 - 2(4) = -1 is odd. It is an even square.
%e A088827 18 is a term since it is even and the sum of its divisors {1,2,3,6,9,18} = 39 - 2(18) = 3 is odd. It is 2 times a square, i.e., 2(9). (End)
%t A088827 Do[s=DivisorSigma[1, n]-2*n; If[OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]
%t A088827 (* Second program: *)
%t A088827 Select[Range[2, 2000, 2], OddQ[DivisorSigma[1, #] - 2 #] &] (* _Michael De Vlieger_, May 14 2017 *)
%o A088827 (Python)
%o A088827 from itertools import count, islice
%o A088827 from sympy.ntheory.primetest import is_square
%o A088827 def A088827_gen(startvalue=2): # generator of terms >= startvalue
%o A088827     return filter(lambda n:is_square(n) or is_square(n>>1),count(max(startvalue+(startvalue&1),2),2))
%o A088827 A088827_list = list(islice(A088827_gen(),30)) # _Chai Wah Wu_, Jul 06 2023
%Y A088827 Cf. A016754, A088828, A088829, A111003, A191432.
%K A088827 nonn,easy
%O A088827 1,1
%A A088827 _Labos Elemer_, Oct 28 2003