A158705 Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.

1, 3, 4, 6, 9, 11, 12, 14, 16, 18, 21, 23, 24, 26, 29, 31, 33, 35, 36, 38, 41, 43, 44, 46, 48, 50, 53, 55, 56, 58, 61, 63, 64, 66, 69, 71, 72, 74, 77, 79, 81, 83, 84, 86, 89, 91, 92, 94, 96, 98, 101, 103, 104, 106, 109, 111, 113, 115, 116, 118, 121, 123, 124, 126

Offset: 1

Author

John W. Layman, Mar 26 2009

Keywords

• nonn
• base

Comments

The nonnegative integers with an even number of even powers of 2 in their base-2 representation are given in A158704.

It appears that a result similar to Prouhet's theorem holds for the terms of A158704 and A158705, specifically: Sum_{k=0..2^n-1, k has an even number of even powers of 2} k^j = Sum_{k=0..2^n-1, k has an odd number of even powers of 2} k^j, for 0 <= j <= (n-1)/2. For a recent treatment of this theorem, see the reference.

Take any binary vector of length 4n+1 with n >= 0. We can activate any bits. When a bit is activated, neighboring bits change their values 0 -> 1, 1 -> 0. Our goal is to turn the original binary vector into a vector of only ones by activating the bits. If the value of the binary vector belongs to this sequence, this is possible for a maximum of 4n+1 activations. - Mikhail Kurkov, Jun 03 2021 [verification needed]

Examples

			The base-2 representation of 6 is 110, i.e., 6 = 2^2 + 2^1, with one even power of 2. Thus 6 is a term of the sequence.
		

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.
• Eric Weisstein's World of Mathematics, Prouhet-Tarry-Escott Problem
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A341389 (characteristic function), A158704 (complement).

Cf. A000069, A001969, A157971.

Programs

Magma

[ n : n in [0..150] | IsOdd(&+Intseq(n, 4))]; // Vincenzo Librandi, Apr 13 2011

Mathematica

Select[Range[100],OddQ[Total[Take[Reverse[IntegerDigits[#,2]],{1,-1,2}]]]&] (* Harvey P. Dale, Dec 23 2012 *)