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A088827
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Even numbers with odd abundance: even squares or two times squares.
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10
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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
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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