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A116882 A number n is included if (highest odd divisor of n)^2 <= n. 27

%I

%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 n is included if (highest odd divisor of n)^2 <= n.

%C Also n is included if (and only if ) the greatest power of 2 dividing n is >= the highest odd divisor of n. 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

%H G. C. Greubel, <a href="/A116882/b116882.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Janjic and B. 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

%e 40 = 8 * 5, where 8 is highest power of 2 dividing 40 and 5 is the highest odd dividing 40. 8 is >= 5 (and, uncoincidently, 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 *)

%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

%Y Cf. A000265, A006519, A080075, A112714.

%Y The complement is A116883.

%Y Positions of 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).

%K nonn,changed

%O 1,2

%A _Leroy Quet_, Feb 24 2006

%E More terms from _Stefan Steinerberger_, Apr 10 2006

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Last modified March 6 20:37 EST 2021. Contains 341850 sequences. (Running on oeis4.)