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A097968 Consider the succession of single digits of the positive even integers: 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 ... (A036211). This sequence is the lexicographically earliest sequence of distinct positive odd integers that produces the same succession of digits.

%I A097968 #31 Jul 09 2017 18:28:30
%S A097968 2468101,21,41,61,82022242628303,23,43,63,84042444648505,25,45,65,
%T A097968 86062646668707,27,47,67,88082848688909,29,49,69,81001021041061081,
%U A097968 101,1,211,411,611,81,201,221,241,261,281,301,3,213,413,613,81401,421,441
%N A097968 Consider the succession of single digits of the positive even integers: 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 ... (A036211). This sequence is the lexicographically earliest sequence of distinct positive odd integers that produces the same succession of digits.
%C A097968 Original name: "Write each odd integer >0 on a single label. Put the labels in numerical order to form an infinite sequence L. Now consider the succession of single digits of A005843 (even numbers): 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8... The sequence S gives a rearrangement of the labels that reproduces the same succession of digits, subject to the constraint that the smallest label must be used that does not lead to a contradiction."
%C A097968 This could be roughly rephrased like this: Rewrite in the most economical way the "even numbers pattern" using only odd numbers, but rearranged. All the numbers of the sequence must be different one from another.
%H A097968 Michael De Vlieger, <a href="/A097968/b097968.txt">Table of n, a(n) for n = 1..10000</a>
%H A097968 Eric Angelini, <a href="http://www.pourlascience.fr/ewb_pages/a/article-jeux-de-suites-19006.php">Jeux de suites</a>, in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
%e A097968 We must begin with "2,4,6..." and we cannot use "2" or "24" or "246" (only odd terms are available), so the first possibility is "2468101". We could not have used "24681" since no term begins with a 0.
%t A097968 f[lst_List, k_] := Block[{L = lst, g, w, a = {}, m}, g[x_] := First@ FirstPosition[x, i_ /; OddQ@ i]; Do[w = Take[L, g@ L]; L = Drop[L, Length@ w]; m = Take[L, g@ L]; While[Or[MemberQ[a, FromDigits@ w], IntegerLength@ FromDigits@ m < Length@ m], w = Join[w, m]; L = Drop[L, Length@ m]; m = Take[L, g@ L]]; AppendTo[a, FromDigits@ w], {k}]; a]; f[Flatten@ Map[IntegerDigits, 2 Range@ 80], 40] (* _Michael De Vlieger_, Nov 28 2015, Version 10 *)
%Y A097968 Cf. A005408, A005843, A098099.
%K A097968 base,easy,nonn
%O A097968 1,1
%A A097968 _Eric Angelini_, Sep 22 2004
%E A097968 Name and Example edited by _Danny Rorabaugh_, Nov 28 2015