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Even numbers with odd abundance: even squares or two times squares.
10

%I #35 Jul 09 2023 03:15:46

%S 2,4,8,16,18,32,36,50,64,72,98,100,128,144,162,196,200,242,256,288,

%T 324,338,392,400,450,484,512,576,578,648,676,722,784,800,882,900,968,

%U 1024,1058,1152,1156,1250,1296,1352,1444,1458,1568,1600,1682,1764,1800,1922

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

%C Sigma(k)-2k is odd means that sigma(k) is also odd.

%C 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 Michael De Vlieger, <a href="/A088827/b088827.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003). - _Amiram Eldar_, Jul 09 2023

%e From _Michael De Vlieger_, May 14 2017: (Start)

%e 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 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 Do[s=DivisorSigma[1, n]-2*n; If[OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]

%t (* Second program: *)

%t Select[Range[2, 2000, 2], OddQ[DivisorSigma[1, #] - 2 #] &] (* _Michael De Vlieger_, May 14 2017 *)

%o (Python)

%o from itertools import count, islice

%o from sympy.ntheory.primetest import is_square

%o def A088827_gen(startvalue=2): # generator of terms >= startvalue

%o return filter(lambda n:is_square(n) or is_square(n>>1),count(max(startvalue+(startvalue&1),2),2))

%o A088827_list = list(islice(A088827_gen(),30)) # _Chai Wah Wu_, Jul 06 2023

%Y Cf. A016754, A088828, A088829, A111003, A191432.

%K nonn,easy

%O 1,1

%A _Labos Elemer_, Oct 28 2003