A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.

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

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(n-1))+2, where t(n) is the n-th 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.

Cf. A000217.

Sequence in context: A219159 A213928 A065189 * A371192 A163363 A065258

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