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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 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

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