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

1,1

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.

Links

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

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

Example

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A226221 A072462 A369951 * A316900 A076057 A364061

Adjacent sequences: A088824 A088825 A088826 * A088828 A088829 A088830

Keyword

nonn,easy

Author

Labos Elemer, Oct 28 2003

Status

approved