A091072 - OEIS

A091072

Positive numbers k such that the Kronecker Symbol (-1 / k) > 0.

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 25, 26, 29, 32, 33, 34, 36, 37, 40, 41, 42, 45, 49, 50, 52, 53, 57, 58, 61, 64, 65, 66, 68, 69, 72, 73, 74, 77, 80, 81, 82, 84, 85, 89, 90, 93, 97, 98, 100, 101, 104, 105, 106, 109, 113, 114, 116, 117, 121, 122, 125, 128, 129

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OFFSET

1,2

COMMENTS

Numbers whose odd part is of the form 4k+1. The bit to the left of the least significant bit of each term is unset. Either of form 2a(m) or 4k+1, k >= 0, 0 < m < n.

A000265(a(n)) is an element of A016813.

a(n) such that A038189(a(n)) = 0.

Numbers n such that kronecker(n, m) = kronecker(m, n) for all m. - Michael Somos, Sep 24 2005

The Dragon curve A014577 (but changing the offset to 1): (1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...) = the characteristic function of A091072. - Gary W. Adamson, Apr 11 2010

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

The terms in the sequence are the same as the terms in the odd columns of the table in A135764 with headings 4k+1: (1, 5, 9, 13...). A014577(n) = 1 if n is in that set, but A014577(n) = 0 if n is in the set of even columns in the A135764 table. - Gary W. Adamson, May 29 2021

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Sep 14 2024

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

J.-P. Allouche, G.-N. Han and J. Shallit, On some conjectures of P. Barry, arXiv:2006.08909 [math.NT], 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 TurnLeft, with a(n) = TurnLeft(n-1).

J. E. S. Socolar and J. M. Taylor, An aperiodic hexagonal tile, arXiv:1003.4279 [math.CO], 2010.

FORMULA

A014707(a(n) + 1) = 0. - Reinhard Zumkeller, Sep 28 2011

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

EXAMPLE

x + 2*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 13*x^8 + 16*x^9 + ...

MAPLE

KS := (n, k) -> NumberTheory:-KroneckerSymbol(n, k):

aList := upto -> select(n -> 0 < KS(-1, n), [seq(1..upto)]):

aList(129); # Peter Luschny, Mar 20 2025

MATHEMATICA

Select[ Range[129], EvenQ[ (#/2^IntegerExponent[#, 2] - 1)/2 ] & ] (* Jean-François Alcover, Feb 16 2012, after Pari *)

PROG

(PARI) for(n=1, 200, if(((n/2^valuation(n, 2)-1)/2)%2==0, print1(n", ")))

(PARI) {a(n) = local(m, c); if( n<1, 0, c=1; m=1; while( c<n, m++; if( ((m / 2^valuation( m, 2) - 1) / 2)%2==0, c++)); m)} /* Michael Somos, Sep 24 2005 */

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

(PARI) isok(k) = kronecker(-1, k) > 0; \\ Michel Marcus, Mar 20 2025

(Haskell)

import Data.List (elemIndices)

a091072 n = a091072_list !! (n-1)

a091072_list = map (+ 1) $ elemIndices 0 a014707_list

-- Reinhard Zumkeller, Sep 28 2011

CROSSREFS

Complement of A091067.

Cf. A000265, A014577 (characteristic function), A014707, A016813, A034947, A055975, A106841 (first of triplet), A088742 (first differences), A339597.

Sequence in context: A393572 A121996 A269176 * A396006 A382092 A001481

Adjacent sequences: A091069 A091070 A091071 * A091073 A091074 A091075