

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
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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


EXAMPLE

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))


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



