A369901 -id:A369901 - cp's OEIS frontend

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.

3, 7, 5, 11, 13, 15, 21, 9, 19, 29, 25, 23, 37, 41, 27, 45, 57, 31, 53, 73, 17, 35, 61, 89, 49, 39, 69, 105, 81, 43, 77, 121, 113, 47, 85, 137, 145, 51, 93, 153, 177, 55, 101, 169, 209, 59, 109, 185, 241, 63, 117, 201, 273, 33, 67, 125, 217, 305, 97

Offset: 1

Thomas Ordowski, Aug 13 2025

A term t must have m = A007814(t-1), and k follows from that so that the representation is unique.
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.
The terms where m is the maximum for each k are A369901 (in order) and are a permutation of the Proth numbers A080075.

Amiram Eldar, Table of n, a(n) for n = 1..11703 (terms corresponding to k < 2500)

Cf. A080075 (Proth numbers in ascending order), A369901 (permutation of Proth numbers).

Table[(k - 2^m)*2^m + 1,{k, 3, 35, 2}, {m, 1, Log2[k-1]}] // Flatten (* Amiram Eldar, Aug 13 2025 *)

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.

More terms from Amiram Eldar, Aug 13 2025

cp's OEIS Frontend

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.

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