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A262356 a(1) = 1; for n > 1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest number not yet present which starts with s, omitting leading zeros.

%I A262356 #22 Mar 12 2019 07:19:40
%S A262356 1,2,3,4,5,6,7,8,9,10,11,12,20,13,30,14,40,15,50,16,60,17,70,18,80,19,
%T A262356 90,21,100,22,23,31,101,102,24,41,103,32,25,51,104,42,26,61,105,52,27,
%U A262356 71,106,62,28,81,107,72,29,91,108,82,200,33,34,43,35,53
%N A262356 a(1) = 1; for n > 1, let s denote the digit-string of a(n-1) with the first digit omitted. Then a(n) is the smallest number not yet present which starts with s, omitting leading zeros.
%C A262356 A simplified variation of A262282.
%C A262356 A permutation of the positive integers with inverse A262358;
%C A262356 A262363 and A262371 give the primes and where they occur: A262363(n)=a(A262371(n)).
%C A262356 a(A262393(n)) = A262390(n).
%C A262356 It seems clear that every number will appear, but it would be nice to have a formal proof. - _N. J. A. Sloane_, Sep 20 2015
%H A262356 Reinhard Zumkeller, <a href="/A262356/b262356.txt">Table of n, a(n) for n = 1..10000</a>
%H A262356 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%t A262356 a[1] = 1; a[n_] := a[n] = Module[{s, k}, s = Rest[IntegerDigits[a[n - 1]]] //. {(0).., d__} :> {d}; For[k = 2, True, k++, If[FreeQ[Array[a, n - 1], k], If[s == {0}, Return[k], If[IntegerDigits[k][[1 ;; Length[s]]] == s, Return[k]]]]]];
%t A262356 Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 12 2019 *)
%o A262356 (Haskell)
%o A262356 import Data.List (isPrefixOf, delete, genericIndex)
%o A262356 import Data.Set (singleton, notMember, insert)
%o A262356 a262356 n = a262356_list !! (n-1)
%o A262356 a262356_list = 1 : f "" (singleton "1") where
%o A262356    f xs s = (read ys :: Int) : f (dropWhile (== '0') ys') (insert ys s)
%o A262356      where ys@(_:ys') = head
%o A262356              [vs | vs <- zss, isPrefixOf xs vs, notMember vs s]
%o A262356    zss = map show [2..]
%Y A262356 Cf. A262283, A262282, A262358 (inverse), A262360 (fixed points), A262374 (binary counterpart), A262363 (primes), A262371, A000030, A262390 (starting with 1), A262393.
%K A262356 nonn,base,look
%O A262356 1,2
%A A262356 _Reinhard Zumkeller_, Sep 19 2015