

A158705


Nonnegative integers with an odd number of even powers of 2 in their base2 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
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OFFSET

1,2


COMMENTS

The nonnegative integers with an even number of even powers of 2 in their base2 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<=(n1)/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. 5762.
Eric Weisstein's World of Mathematics, ProuhetTarryEscott Problem


EXAMPLE

The base2 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



