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A028982
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Squares and twice squares.
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150
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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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
Number of odd divisors of n (A001227) is odd. - Vladeta Jovovic, Aug 28 2007
A071324(a(n)) is odd. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152677(n). - Jaroslav Krizek, Oct 06 2009
Numbers n such that sum of odd divisors of n (A000593) is odd. - Omar E. Pol, Jul 05 2016
A187793(a(n)) is odd. - Timothy L. Tiffin, Jul 18 2016
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
Numbers n such that A175317(n) = Sum_{d|n} pod(d) is odd, where pod(m) = the product of divisors of m (A007955). - Jaroslav Krizek, Dec 28 2016
Positions of zeros in A292377 and A292383, positions of ones in A286357 and A292583. (See A292583 for why.) - Antti Karttunen, Sep 25 2017
Numbers of the form A000079(i)*A016754(j), i,j>=0. - R. J. Mathar, May 30 2020
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
Numbers m whose abundance, A033880(m), is odd. - Peter Munn, May 23 2022
Numbers with an odd number of middle divisors (cf. A067742). - Omar E. Pol, Aug 02 2022
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.
Patrick De Geest, World!Of Numbers
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163.
Eric Weisstein's World of Mathematics, Abundance
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FORMULA
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A001105 UNION A000290.
a(n) is asymptotic to c*n^2 with c = 2/(1+sqrt(2))^2 = 0.3431457.... - Benoit Cloitre, Sep 17 2002
In particular, a(n) = c*n^2 + O(n). - Charles R Greathouse IV, Jan 11 2013
a(A003152(n)) = n^2; a(A003151(n)) = 2*n^2. - Enrique Pérez Herrero, Oct 09 2013
Sum_{n>=1} 1/a(n) = Pi^2/4. - Amiram Eldar, Jun 28 2020
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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
-- Reinhard Zumkeller, Jun 27 2015
(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)))
A028982_list = list(islice(A028982_gen(), 30)) # Chai Wah Wu, Jan 09 2023
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CROSSREFS
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Complement of A028983.
Characteristic function is A053866, A093709.
Odd terms in A178910.
Cf. A000203, A000290, A000593, A001105, A042968, A187793.
Supersequence of A000079.
Cf. A028260, A033880, A046951, A067742.
Cf. A119620, A119899, A347437, A347438, A348550.
Sequence in context: A336232 A242662 A335851 * A320137 A324525 A175338
Adjacent sequences: A028979 A028980 A028981 * A028983 A028984 A028985
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KEYWORD
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nonn,easy
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AUTHOR
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Patrick De Geest
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STATUS
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approved
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