

A165275


Table read by antidiagonals: T(n, k) is the kth number with n1 oddpower summands in its base 2 representation.


1



1, 4, 2, 5, 3, 10, 16, 6, 11, 42, 17, 7, 14, 43, 170, 20, 8, 15, 46, 171, 682, 21, 9, 26, 47, 174, 683, 2730, 64, 12, 27, 58, 175, 686, 2731, 10922, 65, 13, 30, 59, 186, 687, 2734, 10923, 43690, 68, 18, 31, 62, 187, 698, 2735, 10926, 43691, 174762, 69, 19, 34
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OFFSET

1,2


COMMENTS

For n>=0, row n is the ordered sequence of positive integers m such that the number of odd powers of 2 in the base 2 representation of m is n.
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For even powers, see A165274. For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
It appears that, for n>=3, a(t(n)) = 4*a(t(n1))+2, where t(n) is the nth triangular number t(n)=n(n+1)/2 (A000217). [John W. Layman, Sep 15 2009]


LINKS

Table of n, a(n) for n=1..58.


EXAMPLE

Northwest corner:
1....4....5...16...17...20...21...64
2....3....6....7....8....9...12...13
10..11...14...26...27...30...31...34
42..43...46...47...58...59...62...63
Examples:
20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.


MATHEMATICA

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[2 ;; 1 ;; 2]]]; T = GatherBy[ SortBy[Range[10^5], f], f]; Table[Table[T[[n  k + 1, k]], {k, n, 1, 1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020*)


CROSSREFS

Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.
A000217 [From John W. Layman, Sep 15 2009]
Sequence in context: A219159 A213928 A065189 * A163363 A065258 A049259
Adjacent sequences: A165272 A165273 A165274 * A165276 A165277 A165278


KEYWORD

nonn,tabl


AUTHOR

Clark Kimberling, Sep 12 2009


EXTENSIONS

a(27) corrected and a(28)a(54) added by John W. Layman, Sep 15 2009
More terms from Amiram Eldar, Feb 04 2020


STATUS

approved



