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%I #8 Feb 04 2020 08:13:19
%S 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,
%T 93,343,1365,128,12,19,53,95,349,1367,5461,130,14,20,55,117,351,1373,
%U 5463,21845,136,16,22,61,119,373,1375,5469,21847,87381,138,18,25,63
%N Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-power summands in its base 2 representation.
%C 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.
%C Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
%C For odd powers, see A165275.
%C For the number of even powers of 2 in the base 2 representation of n, see A139351; for odd, see A139352.
%C Essentially, (Row 0)=A062880, (Row 1)=A158705, (Column 1)=A002450, also possibly (Column 2)=A163832.
%e Northwest corner:
%e 2....8...10...32...34...40...42...129
%e 1....3....4....6....9...11...12...14
%e 5....7...13...15...17...19...20...22
%e 21..23...29...31...53...55...61...63
%e Examples:
%e 40 = 32 + 8 = 2^5 + 2^3, so that 40 is in row 0.
%e 13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 2.
%t 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*)
%Y Cf. A139351, A139352, A165275, A165276, A165277, A165278, A165279.
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Sep 12 2009
%E More terms from _Amiram Eldar_, Feb 04 2020
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