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A387016 Permutation of the odd integers >= 3 formed by ordering them first by odd k >= 3 and then by integer m >= 1 in their unique representation (k - 2^m)*2^m + 1.

%I A387016 #74 Aug 22 2025 08:45:20
%S A387016 3,7,5,11,13,15,21,9,19,29,25,23,37,41,27,45,57,31,53,73,17,35,61,89,
%T A387016 49,39,69,105,81,43,77,121,113,47,85,137,145,51,93,153,177,55,101,169,
%U A387016 209,59,109,185,241,63,117,201,273,33,67,125,217,305,97
%N A387016 Permutation of the odd integers >= 3 formed by ordering them first by odd k >= 3 and then by integer m >= 1 in their unique representation (k - 2^m)*2^m + 1.
%C A387016 A term t must have m = A007814(t-1), and k follows from that so that the representation is unique.
%C A387016 For given k, successive terms have m in the range 1 <= m <= floor(log_2(k)) and this regularity permits a(n) to be calculated from the index n.
%C A387016 The terms where m is the maximum for each k are A369901 (in order) and are a permutation of the Proth numbers A080075.
%H A387016 Amiram Eldar, <a href="/A387016/b387016.txt">Table of n, a(n) for n = 1..11703</a> (terms corresponding to k < 2500)
%F A387016 P(n,m) = (2n+1 - 2^m)*2^m + 1 = (2n+1)*2^m - 4^m + 1, where m > 0 with 2^m < 2n+1, for n > 0.
%t A387016 Table[(k - 2^m)*2^m + 1,{k, 3, 35, 2}, {m, 1, Log2[k-1]}] // Flatten (* _Amiram Eldar_, Aug 13 2025 *)
%Y A387016 Cf. A080075 (Proth numbers in ascending order), A369901 (permutation of Proth numbers).
%K A387016 nonn,easy,look,new
%O A387016 1,1
%A A387016 _Thomas Ordowski_, Aug 13 2025
%E A387016 More terms from _Amiram Eldar_, Aug 13 2025