A307620 Integers x such that [f(0), f(f(0)), ..., f(...f(0)...)] is a permutation of [0, 1, ..., k-1], where k is the number of digits in x and f(a) denotes the 0-based index of the first occurrence of the substring a in x.

0, 10, 120, 201, 1230, 1302, 2031, 2310, 3012, 3201, 12340, 12403, 13042, 13420, 14023, 14302, 20341, 20413, 23140, 23401, 24103, 24310, 30142, 30421, 32041, 32410, 34012, 34120, 40123, 40312, 42013, 42301, 43021, 43102

Offset

1,2

Comments

Numbers with 21 digits cannot fit every number from 0 to 20 as strings. Hence this sequence cannot have any numbers with 21 digits or more, making it finite.

1918171615141312110 is the largest number in this sequence (see examples).

Links

Table of n, a(n) for n=1..34.

Example

Consider x = 40312. f(0) = 1, f(1) = 3, f(3) = 2, f(2) = 4, f(4) = 0 and so we have visited every index of x.
Consider the largest number in this sequence, x = 1918171615141312110. f(0) = 18, f(18) = 2, f(2) = 15, f(15) = 8, f(8) = 3, f(3) = 13, f(13) = 12, f(12) = 14, f(14) = 10, f(10) = 17, f(17) = 4, f(4) = 11, f(11) = 16, f(16) = 6, f(6) = 7, f(7) = 5, f(5) = 9, f(9) = 1, f(1) = 0 and so we have visited every index of x.

Mathematica

{{0}}~Join~Array[Block[{k = #, r}, r = Range[0, k]; FromDigits /@ Select[DeleteCases[Permutations[r, {k + 1}], _?(First@ # == 0 &)], With[{w = #}, Union@ Rest@ Nest[Append[#, Position[w, #[[-1]]][[1, 1]] - 1] &, {0}, Length@ w] == r] &]] &, 4] // Flatten (* Michael De Vlieger, Apr 21 2019 *)

Crossrefs

Sequence in context: A373711 A069498 A226767 * A300522 A252874 A182605

Adjacent sequences: A307617 A307618 A307619 * A307621 A307622 A307623

Keyword

nonn,fini,base

Author

Dmitry Kamenetsky, Apr 18 2019

Status

approved