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A165274
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Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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 the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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*)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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