cp's OEIS Frontend

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

%I A165275 #13 Feb 25 2023 06:06:50
%S A165275 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,
%T A165275 174,683,2730,64,12,27,58,175,686,2731,10922,65,13,30,59,186,687,2734,
%U A165275 10923,43690,68,18,31,62,187,698,2735,10926,43691,174762,69,19,34
%N A165275 Table read by antidiagonals: T(n, k) is the k-th number with n-1 odd-power summands in its base 2 representation.
%C A165275 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.
%C A165275 Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
%C A165275 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.
%C A165275 Essentially, (Row 0)=A000695, (Column 1)=A020988, also possibly (Column 2)=A007583.
%C A165275 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]
%e A165275 Northwest corner:
%e A165275   1....4....5...16...17...20...21...64
%e A165275   2....3....6....7....8....9...12...13
%e A165275   10..11...14...26...27...30...31...34
%e A165275   42..43...46...47...58...59...62...63
%e A165275 Examples:
%e A165275 20 = 16 + 4 = 2^4 + 2^2, so that 20 is in row 0.
%e A165275 13 = 8 + 4 + 1 = 2^3 + 2^2 + 2^0, so that 13 is in row 1.
%t A165275 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*)
%Y A165275 Cf. A139351, A139352, A165274, A165276, A165277, A165278, A165279.
%Y A165275 Cf. A000217.
%K A165275 nonn,tabl
%O A165275 1,2
%A A165275 _Clark Kimberling_, Sep 12 2009
%E A165275 a(27) corrected and a(28)-a(54) added by _John W. Layman_, Sep 15 2009
%E A165275 More terms from _Amiram Eldar_, Feb 04 2020