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A158705
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Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.
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4
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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
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range[100], OddQ[Total[Take[Reverse[IntegerDigits[#, 2]], {1, -1, 2}]]]&] (* Harvey P. Dale, Dec 23 2012 *)
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PROG
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(Magma) [ n : n in [0..150] | IsOdd(&+Intseq(n, 4))]; // Vincenzo Librandi, Apr 13 2011
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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