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A088828
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Nonsquare positive odd numbers.
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13
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3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139
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OFFSET
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1,1
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COMMENTS
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Odd numbers with even abundance: primes and some composites too.
Odd numbers with odd abundance are in A016754. Even numbers with odd abundance are in A088827. Even numbers with even abundance are in A088829.
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LINKS
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FORMULA
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a(n) = 2*n + s - ((s+1) mod 2) where s = round(sqrt(2*n-1)). - Gerald Hillier, Apr 15 2009
a(n) = 2*(n+h) + 1 where h = floor((1/4)*(sqrt(8*n) - 1)) is the largest value such that A014105(h) < n. - John Tyler Rascoe, Jul 05 2022
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EXAMPLE
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n = p prime, abundance = 1 - p = even and negative;
n = 21, sigma = 1 + 3 + 7 + 21 = 32, abundance = 32 - 42 = -20.
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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}]
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PROG
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(Magma) [ n: n in [1..140 by 2] | IsEven(SumOfDivisors(n)-2*n) ]; // Klaus Brockhaus, Apr 15 2009
(PARI) isok(n) = (n>0) && (n % 2) && ! issquare(n); \\ Michel Marcus, Aug 28 2013
(Python)
from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A088828_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:not is_square(n), count(max(startvalue+(startvalue&1^1), 1), 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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EXTENSIONS
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STATUS
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approved
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