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

1,2

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

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).

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A330684 A070173 A363122 * A069519 A363948 A087980

Adjacent sequences: A116879 A116880 A116881 * A116883 A116884 A116885

Keyword

nonn,easy

Author

Leroy Quet, Feb 24 2006

Extensions

More terms from Stefan Steinerberger, Apr 10 2006

Status

approved