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

1, 2, 4, 8, 12, 16, 24, 32, 40, 48, 56, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1088, 1152, 1216, 1280, 1344, 1408

Offset: 1

Author

Leroy Quet, Feb 24 2006

Keywords

• nonn
• easy

Comments

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.

Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - Thomas Ordowski, Oct 19 2014

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

Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - Amiram Eldar, Jan 24 2023

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).

Crossrefs

Cf. A000265, A006519, A080075, A112714.

The complement is A116883.

Positions of zeros (and 1) in A341675.

A051283 = numbers without a superior prime-power divisor (zeros of A341593).

A059172 = numbers without a superior squarefree divisor (zeros of A341592).

A063539 = numbers without a superior prime divisor (zeros of A341591).

A333805 counts strictly inferior odd divisors.

A341594 counts strictly superior odd divisors.

- Inferior: A033676, A038548, A063962, A066839, A069288, A161906, A217581.

- Superior: A033677, A063538, A070038, A072500, A161908, A341676.

- Strictly Inferior: A056924, A060775, A070039, A333806, A341596, A341674.

- Strictly Superior: A056924, A064052/A048098, A140271, A238535, A341642, A341643, A341673.

Cf. A000005, A000203, A001248, A006530, A020639, A026804, A027193, A340101, A340854 (zeros of A340832), A001008, A002805.

Subsequence of A082662, {1} U A363122.

Programs

Mathematica

f[n_] := Select[Divisors[n], OddQ[ # ] &][[ -1]]; Insert[Select[Range[2, 1500], 2^FactorInteger[ # ][[1]][[2]] > f[ # ] &], 1, 1] (* Stefan Steinerberger, Apr 10 2006 *)
q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Select[Range[1500], q] (* Amiram Eldar, Jan 24 2023 *)

PARI

isok(n) = vecmax(select(x->((x % 2)==1), divisors(n)))^2 <= n; \\ Michel Marcus, Sep 06 2016

PARI

isok(n) = 2^(valuation(n,2)*2) >= n \\ Jeppe Stig Nielsen, Feb 19 2019

Python

from itertools import count, islice
def A116882_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n)**2>=n,count(max(startvalue,1)))
A116882_list = list(islice(A116882_gen(),20)) # Chai Wah Wu, May 17 2023

Formula

a(n) = A080075(n-1)-1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010

a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014

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

Extensions

More terms from Stefan Steinerberger, Apr 10 2006