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A088827 Even numbers with odd abundance: even squares or two times squares. 10
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A226221 A072462 A369951 * A316900 A076057 A364061
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Oct 28 2003
STATUS
approved

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Last modified February 26 12:54 EST 2024. Contains 370352 sequences. (Running on oeis4.)