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

2, 4, 8, 16, 18, 32, 36, 50, 64, 72, 98, 100, 128, 144, 162, 196, 200, 242, 256, 288, 324, 338, 392, 400, 450, 484, 512, 576, 578, 648, 676, 722, 784, 800, 882, 900, 968, 1024, 1058, 1152, 1156, 1250, 1296, 1352, 1444, 1458, 1568, 1600, 1682, 1764, 1800, 1922

Offset: 1

Author

Labos Elemer, Oct 28 2003

Keywords

• nonn
• easy

Comments

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

Odd numbers with odd abundance are in A016754. Odd numbers with even abundance are in A088828. Even numbers with even abundance are in A088829.

Examples

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

Do[s=DivisorSigma[1, n]-2*n; If[OddQ[s]&&!OddQ[n], Print[{n, s}]], {n, 1, 1000}]
(* Second program: *)
Select[Range[2, 2000, 2], OddQ[DivisorSigma[1, #] - 2 #] &] (* Michael De Vlieger, May 14 2017 *)

Python

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A088827_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:is_square(n) or is_square(n>>1),count(max(startvalue+(startvalue&1),2),2))
A088827_list = list(islice(A088827_gen(),30)) # Chai Wah Wu, Jul 06 2023

Formula

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

Sum_{n>=1} 1/a(n) = Pi^2/8 (A111003). - Amiram Eldar, Jul 09 2023