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%I #68 Oct 04 2023 06:48:59
%S 1,2,4,8,12,16,24,32,40,48,56,64,80,96,112,128,144,160,176,192,208,
%T 224,240,256,288,320,352,384,416,448,480,512,544,576,608,640,672,704,
%U 736,768,800,832,864,896,928,960,992,1024,1088,1152,1216,1280,1344,1408
%N A number k is included if (highest odd divisor of k)^2 <= k.
%C Also k is included if (and only if) the greatest power of 2 dividing k is >= the highest odd divisor of k. All terms of the sequence are even besides the 1.
%C Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - _Thomas Ordowski_, Oct 19 2014
%C If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence consists of 1 and all numbers without a superior odd divisor. - _Gus Wiseman_, Feb 18 2021
%C Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - _Amiram Eldar_, Jan 24 2023
%H G. C. Greubel, <a href="/A116882/b116882.txt">Table of n, a(n) for n = 1..1000</a>
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 38.
%H Milan Janjic and Boris Petkovic, <a href="http://arxiv.org/abs/1301.4550">A Counting Function</a>, arXiv 1301.4550 [math.CO], 2013.
%H Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3339313">Discussion of the groupoid of Proth numbers (OEIS A080075)</a>, Politecnico di Torino, Italy (2019).
%F a(n) = A080075(n-1)-1. - _Klaus Brockhaus_, _Georgi Guninski_ and _M. F. Hasler_, Aug 16 2010
%F a(n) ~ n^2/2. - _Thomas Ordowski_, Oct 19 2014
%F Sum_{n>=1} 1/a(n) = 1 + (3/4) * Sum_{k>=1} H(2^k-1)/2^k = 2.3388865091..., where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - _Amiram Eldar_, Jan 24 2023
%e 40 = 8 * 5, where 8 is highest power of 2 dividing 40 and 5 is the highest odd dividing 40. 8 is >= 5 (so 5^2 <= 40), so 40 is in the sequence.
%t f[n_] := Select[Divisors[n], OddQ[ # ] &][[ -1]]; Insert[Select[Range[2, 1500], 2^FactorInteger[ # ][[1]][[2]] > f[ # ] &], 1, 1] (* _Stefan Steinerberger_, Apr 10 2006 *)
%t q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Select[Range[1500], q] (* _Amiram Eldar_, Jan 24 2023 *)
%o (PARI) isok(n) = vecmax(select(x->((x % 2)==1), divisors(n)))^2 <= n; \\ _Michel Marcus_, Sep 06 2016
%o (PARI) isok(n) = 2^(valuation(n,2)*2) >= n \\ _Jeppe Stig Nielsen_, Feb 19 2019
%o (Python)
%o from itertools import count, islice
%o def A116882_gen(startvalue=1): # generator of terms >= startvalue
%o return filter(lambda n:(n&-n)**2>=n,count(max(startvalue,1)))
%o A116882_list = list(islice(A116882_gen(),20)) # _Chai Wah Wu_, May 17 2023
%Y Cf. A000265, A006519, A080075, A112714.
%Y The complement is A116883.
%Y Positions of zeros (and 1) in A341675.
%Y A051283 = numbers without a superior prime-power divisor (zeros of A341593).
%Y A059172 = numbers without a superior squarefree divisor (zeros of A341592).
%Y A063539 = numbers without a superior prime divisor (zeros of A341591).
%Y A333805 counts strictly inferior odd divisors.
%Y A341594 counts strictly superior odd divisors.
%Y - Inferior: A033676, A038548, A063962, A066839, A069288, A161906, A217581.
%Y - Superior: A033677, A063538, A070038, A072500, A161908, A341676.
%Y - Strictly Inferior: A056924, A060775, A070039, A333806, A341596, A341674.
%Y - Strictly Superior: A056924, A064052/A048098, A140271, A238535, A341642, A341643, A341673.
%Y Cf. A000005, A000203, A001248, A006530, A020639, A026804, A027193, A340101, A340854 (zeros of A340832), A001008, A002805.
%Y Subsequence of A082662, {1} U A363122.
%K nonn,easy
%O 1,2
%A _Leroy Quet_, Feb 24 2006
%E More terms from _Stefan Steinerberger_, Apr 10 2006
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