A091067 Numbers whose odd part is of the form 4k+3.

3, 6, 7, 11, 12, 14, 15, 19, 22, 23, 24, 27, 28, 30, 31, 35, 38, 39, 43, 44, 46, 47, 48, 51, 54, 55, 56, 59, 60, 62, 63, 67, 70, 71, 75, 76, 78, 79, 83, 86, 87, 88, 91, 92, 94, 95, 96, 99, 102, 103, 107, 108, 110, 111, 112, 115, 118, 119, 120, 123, 124, 126, 127, 131

Offset: 1

Author

Ralf Stephan, Feb 22 2004

Keywords

• nonn
• easy

Comments

Either of form 2*a(m) or 4k+3, k >= 0, 0 < m < n.

A000265(a(n)) is an element of A004767.

a(n) such that A038189(a(n)) = 1.

Numbers n such that Kronecker(-n, m) = Kronecker(m, n) for all m. - Michael Somos, Sep 22 2005

From Antti Karttunen, Feb 20-21 2015: (Start)

Gives all n for which A005811(n) - A005811(n-1) = -1, from which follows that a(n) = the least k such that A255070(k) = n.

Gives the positions of even terms in A003602. (End)

Indices of negative terms in A164677. - M. F. Hasler, Aug 06 2015

Indices of the 0's in A014577. - Gabriele Fici, Jun 02 2016

Also indices of -1 in A034947. - Jianing Song, Apr 24 2021

Conjecture: alternate definition of same sequence is that a(1)=3 and a(n) is the smallest number > a(n-1) so that no number that is the sum of at most 2 terms in this sequence is a power of 2. - J. Lowell, Jan 20 2024

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Aug 31 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
• Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
• Kevin Ryde, Iterations of the Dragon Curve, see index TurnRight, with a(n) = TurnRight(n-1).

Crossrefs

Essentially one less than A060833.

Characteristic function: A038189.

Complement of A091072.

First differences are in A106836 (from its second term onward).

Sequence A246590 gives the even terms.

Gives the positions of records (after zero) for A255070 (equally, the position of the first n there).

Cf. A106837 (gives n such that both n and n+1 are terms of this sequence).

Cf. A098502 (gives n such that both n and n+2 are, but n+1 is not in this sequence).

Cf. also A000265, A003602, A004767, A005811, A014707, A034947, A055975, A236840, A255068, A255327, A255330.

Programs

Haskell

import Data.List (elemIndices)
a091067 n = a091067_list !! (n-1)
a091067_list = map (+ 1) $ elemIndices 1 a014707_list
-- Reinhard Zumkeller, Sep 28 2011
(Scheme, with Antti Karttunen's IntSeq-library, two versions)
(define A091067 (MATCHING-POS 1 1 (COMPOSE even? A003602)))
(define A091067 (NONZERO-POS 1 0 A038189))
;; Antti Karttunen, Feb 20 2015

Mathematica

Select[Range[150], Mod[# / 2^IntegerExponent[#, 2], 4] == 3 &] (* Amiram Eldar, Aug 31 2024 *)

PARI

for(n=1,200,if(((n/2^valuation(n,2)-1)/2)%2,print1(n",")))

PARI

{a(n) = local(m, c); if( n<1, 0, c=0; m=1; while( cMichael Somos, Sep 22 2005 */

PARI

is_A091067(n)=bittest(n,valuation(n,2)+1) \\ M. F. Hasler, Aug 06 2015

PARI

a(n) = my(t=1); n<<=1; forstep(i=logint(n,2),0,-1, if(bittest(n,i)==t, n++;t=!t)); n; \\ Kevin Ryde, Mar 21 2021

Formula

a(n) = A060833(n+1) - 1. [See N. Sato's Feb 12 2013 comment in A060833.]

Other identities. For all n >= 1 it holds that:

A014707(a(n) + 1) = 1. - Reinhard Zumkeller, Sep 28 2011

A055975(a(n)) < 0. - Reinhard Zumkeller, Apr 28 2012

From Antti Karttunen, Feb 20-21 2015: (Start)

a(n) = A246590(n)/2.

A255070(a(n)) = n, or equally, A236840(a(n)) = 2n.

a(n) = 1 + A255068(n-1). (End)