A157971 Odious twin locations: first members of pairs of consecutive odious numbers.

1, 7, 13, 21, 25, 31, 37, 41, 49, 55, 61, 69, 73, 81, 87, 93, 97, 103, 109, 117, 121, 127, 133, 137, 145, 151, 157, 161, 167, 173, 181, 185, 193, 199, 205, 213, 217, 223, 229, 233, 241, 247, 253, 261, 265, 273, 279, 285, 289, 295

Offset: 1

Author

John W. Layman, Mar 10 2009

Keywords

• nonn
• easy

Comments

An odious number (A000069) is a nonnegative integer with an odd number of ones in its binary expansion.

In the reference it is shown that these odious twins alternate with the evil twins (see A157970), which are pairs of consecutive evil numbers (A001969) having even numbers of ones in their binary expansions.

Examples

			The sequence of odious numbers (A000069) begins 1,2,4,7,8,11,13,14,16,19,21,..., so the first few odious twins are at 1,7,13, ... .
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag. 82 (2009), pp. 57-62; also on JSTOR.
• Eric Weisstein's World of Mathematics, Odious Number.

Crossrefs

Cf. A000069, A091855, A157970, A248057.

Programs

Mathematica

SequencePosition[Table[If[OddQ[DigitCount[n,2,1]],1,0],{n,300}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 05 2016 *)

PARI

lista(nn) = select(n->((hammingweight(n) % 2) && (hammingweight(n+1) % 2)), vector(nn, i, i)); \\ Michel Marcus, Jul 10 2014

Formula

a(n) = A248057(n) - 1. - Amiram Eldar, Jun 16 2025

Extensions

Comment corrected by Jeff Aronson. - N. J. A. Sloane, Oct 04 2020