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A158705 Nonnegative integers with an odd number of even powers of 2 in their base-2 representation. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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^j, 0<=k<2^n, k has an even number of even powers of 2]

= Sum[k^j, 0<=k<2^n, k has an odd number of even powers of 2],

for 0<=j<=(n-1)/2. For a recent treatment of this theorem, see the reference.

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

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000069, A001969, A157971, A158704.

Sequence in context: A037969 A329862 A153236 * A047415 A087805 A213040

Adjacent sequences:  A158702 A158703 A158704 * A158706 A158707 A158708

KEYWORD

nonn

AUTHOR

John W. Layman, Mar 26 2009

STATUS

approved

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)