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A028982
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Squares and twice squares.
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158
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1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, 121, 128, 144, 162, 169, 196, 200, 225, 242, 256, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 512, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1024
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OFFSET
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1,2
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COMMENTS
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Numbers n such that sum of divisors of n (A000203) is odd.
Also the numbers with an odd number of run sums (trapezoidal arrangements, number of ways of being written as the difference of two triangular numbers). - Ron Knott, Jan 27 2003
Pell(n)*Sum_{k|n} 1/Pell(k) is odd, where Pell(n) is A000129(n). - Paul Barry, Oct 12 2005
If k is odd (k = 2m+1 for m >= 0), then 2^k = 2^(2m+1) = 2*(2^m)^2. If k is even (k = 2m for m >= 0), then 2^k = 2^(2m) = (2^m)^2. So, the powers of 2 sequence (A000079) is a subsequence of this one. - Timothy L. Tiffin, Jul 18 2016
Equivalently, numbers whose odd part is square. Cf. A042968. - Peter Munn, Jul 14 2020
These are the Heinz numbers of the partitions counted by A119620. - Gus Wiseman, Oct 29 2021
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LINKS
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Eric Weisstein's World of Mathematics, Abundance
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FORMULA
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a(n) is asymptotic to c*n^2 with c = 2/(1+sqrt(2))^2 = 0.3431457.... - Benoit Cloitre, Sep 17 2002
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MATHEMATICA
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Take[ Sort[ Flatten[ Table[{n^2, 2n^2}, {n, 35}] ]], 57] (* Robert G. Wilson v, Aug 27 2004 *)
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PROG
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(PARI) list(lim)=vecsort(concat(vector(sqrtint(lim\1), i, i^2), vector(sqrtint(lim\2), i, 2*i^2))) \\ Charles R Greathouse IV, Jun 16 2011
(Haskell)
import Data.List.Ordered (union)
a028982 n = a028982_list !! (n-1)
a028982_list = tail $ union a000290_list a001105_list
(Python)
from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A028982_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:int(is_square(n) or is_square(n<<1)), count(max(startvalue, 1)))
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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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