A165274 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.

2, 8, 1, 10, 3, 5, 32, 4, 7, 21, 34, 6, 13, 23, 85, 40, 9, 15, 29, 87, 341, 42, 11, 17, 31, 93, 343, 1365, 128, 12, 19, 53, 95, 349, 1367, 5461, 130, 14, 20, 55, 117, 351, 1373, 5463, 21845, 136, 16, 22, 61, 119, 373, 1375, 5469, 21847, 87381, 138, 18, 25, 63

Offset: 1

Author

Clark Kimberling, Sep 12 2009

Keywords

• nonn
• tabl

Comments

For n>=0, row n is the ordered sequence of positive integers m such that the number of even 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 odd powers, see A165275.

For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.

Essentially, (Row 0)=A062880, (Row 1)=A158705, (Column 1)=A002450, also possibly (Column 2)=A163832.

Examples

			Northwest corner:
2....8...10...32...34...40...42...129
1....3....4....6....9...11...12...14
5....7...13...15...17...19...20...22
21..23...29...31...53...55...61...63
Examples:
40 = 32 + 8 = 2^5 + 2^3, so that 40 is in row 0.
13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 2.
		

Crossrefs

Cf. A139351, A139352, A165275, A165276, A165277, A165278, A165279.

Programs

Mathematica

f[n_] := Total[(Reverse@IntegerDigits[n, 2])[[1 ;; -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*)

Extensions

More terms from Amiram Eldar, Feb 04 2020