A158704 Nonnegative integers with an even number of even powers of 2 in their base-2 representation.

0, 2, 5, 7, 8, 10, 13, 15, 17, 19, 20, 22, 25, 27, 28, 30, 32, 34, 37, 39, 40, 42, 45, 47, 49, 51, 52, 54, 57, 59, 60, 62, 65, 67, 68, 70, 73, 75, 76, 78, 80, 82, 85, 87, 88, 90, 93, 95, 97, 99, 100

Offset: 1

Author

John W. Layman, Mar 24 2009

Keywords

• nonn
• base

Comments

The nonnegative integers with an odd number of even powers of 2 in their base-2 representation are given in A158705.

It appears that a result similar to Prouhet's theorem holds for the terms of A158704 and A158705, specifically: Sum_{k=0..2^n-2, k has an even number of even powers of 2} k^j = Sum_{k=0..2^n-2, 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.

Conjecture: take any binary vector of length 4n+3 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+3 activations. - Mikhail Kurkov, Jun 01 2021

Examples

			The base-2 representation of 5 is 101, i.e., 5 = 2^2 + 2^0, with two even powers of 2. Thus 5 is a term of the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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. A112539 (characteristic function, up to offset), A158705 (complement).

Cf. A000069, A001969, A157971.

Programs

Magma

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

Mathematica

Select[Range[0, 100], EvenQ[Total[Take[Reverse[IntegerDigits[#, 2]], {1, -1, 2}]]] &] (* Amiram Eldar, Jan 04 2020 after Harvey P. Dale at A158705 *)