A365278 In the binary expansion of n replace each run of k consecutive 1's by the decimal digits of A007931(k) to get the ternary expansion of a(n).

0, 1, 3, 2, 9, 10, 6, 4, 27, 28, 30, 11, 18, 19, 12, 5, 81, 82, 84, 29, 90, 91, 33, 31, 54, 55, 57, 20, 36, 37, 15, 7, 243, 244, 246, 83, 252, 253, 87, 85, 270, 271, 273, 92, 99, 100, 93, 32, 162, 163, 165, 56, 171, 172, 60, 58, 108, 109, 111, 38, 45, 46, 21

Offset: 0

Author

Rémy Sigrist, Aug 30 2023

Keywords

• nonn
• base

Comments

This sequence is a permutation of the nonnegative integers with inverse A365279.

For any pair (b, c) of bases >= 2, we can devise a similar sequence, say F_{b, c}:

- for any d >= 2, let Z_d be the set of zeroless numbers in base d,

- in the base b expansion of n replace each run of consecutive nonzero digits (say corresponding to Z_b(k) for some k > 0) by the base c digits of Z_c(k) to get the base c expansion of F_{b, c}(n),

- F_{b, c} is a permutation of the nonnegative integers with inverse F_{c, b},

- F_{c, d} o F_{b, c} = F_{b, d} and F_{b, b} is the identity,

- in particular the present sequence corresponds to F_{2, 3} and its inverse to F_{3, 2}.

Examples

			The binary expansion of 415 is "110011111", A007931(2) = 2 and A007931(5) = 21, so the ternary expansion of a(415) is "20021", and a(415) = 169.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, PARI program
• Index entries for sequences related to binary expansion of n
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A007931, A023416, A032924, A077267, A290308, A365279 (inverse).

Programs

Mathematica

A007931[n_]:=Rest[IntegerDigits[n+1,2]]+1;
A365278[n_]:=FromDigits[Flatten[Map[If[First[#]==1,A007931[Length[#]],#]&,Split[IntegerDigits[n,2]]]],3];
Array[A365278,100,0] (* Paolo Xausa, Oct 17 2023 *)

PARI

See Links section.

Formula

a(2*n) = 3*a(n).

a(2^k - 1) = A032924(k) for any k > 0.

A077267(a(n)) = A023416(n).