A386386 Permutation of the nonnegative integers formed by their unique representation (k - 2^m)*2^(m-1) - 1, where 1 < 2^m < k odd, sorted first by k then by m.

0, 2, 1, 4, 5, 6, 9, 3, 8, 13, 11, 10, 17, 19, 12, 21, 27, 14, 25, 35, 7, 16, 29, 43, 23, 18, 33, 51, 39, 20, 37, 59, 55, 22, 41, 67, 71, 24, 45, 75, 87, 26, 49, 83, 103, 28, 53, 91, 119, 30, 57, 99, 135, 15, 32, 61, 107, 151, 47, 34, 65, 115, 167, 79, 36, 69, 123, 183, 111

Offset

0,2

Comments

It is provable that every nonnegative integer can be written uniquely in the form (k - 2^m)*2^(m-1) - 1, where 1 < 2^m < k odd. Cf. A387016 (with a similar formula), where such uniqueness is proven. Similarly here, a term t must have m = A007814(2t+2), and k follows from that so that the representation is unique.

This allows us to determine the consecutive terms a(n) of this sequence that is a permutation of all integers >= 0 based on A387016 created similarly (see here and there the first and second formulas).

By the expressions k = 2*A030530(n+1) + 1 and m = n - A001855(A030530(n+1)), we obtain an explicit formula for a(n); i.e., dependent only on its index n. Let us also note that k = A386220(a(n-1)+1), by inversion.

Even numbers appear in ascending order at a(A001855(n)) = 2*n; the first term of each k (i.e., with m = 1).

Links

Amiram Eldar, Table of n, a(n) for n = 0..11702 (terms corresponding to k < 2500)

Index entries for sequences that are permutations of the natural numbers.

Formula

a(n) = (A387016(n+1) - 1)/2 - 1.

a(n) = (k-2^m)*2^(m-1) - 1 = k*2^(m-1)-2^(2m-1)-1 where n = A001855((k-1)/2) + m - 1 with 1 < 2^m < k odd.

a(n) = A089654(n+1)*(2*A030530(n+1) - A089654(n+1) + 1)/2 - 1.

Example

Initial terms and their k and m begin
  n    = 0   1   2   3   4   5   6   7   8   9  10  ...
  a(n) = 0,  2,  1,  4,  5,  6,  9,  3,  8, 13, 11, ...
  k    = 3   5   5   7   7   9   9   9  11  11  11
  m    = 1   1   2   1   2   1   2   3   1   2   3
  runs       \---/   \---/   \-------/   \-------/
For odd k=7 and m=2, we have 1<2^2<7, so a(n)=(7-2^2)*2^(2-1)-1=5 where n=A001855((7-1)/2)+2-1=3+1=4.
For odd k=9 and m=1, we have 1<2^1<9, so a(n)=(9-2^1)*2^(1-1)-1=6 where n=A001855((9-1)/2)+1-1=5+0=5.

Mathematica

Table[(k-2^m)*2^(m-1)-1, {k, 3, 50, 2}, {m, 1, Log2[k]}] // Flatten (* Amiram Eldar, Aug 21 2025 *)

Crossrefs

Cf. A001477, A001855, A007814 (2-adic valuation), A030530, A089654, A386220, A387016.

Sequence in context: A261608 A351250 A351253 * A110332 A052947 A159287

Adjacent sequences: A386383 A386384 A386385 * A386387 A386388 A386389

Keyword

nonn,look

Author

Thomas Ordowski, Aug 20 2025

Extensions

More terms from Michel Marcus, Aug 20 2025

Status

approved