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Answering this question, another question came to my mind. For which $n$ will the greedy algorithm work?

We define a sequence of natural numbers $x_n$ recursively: $$x_1 =1,$$ $$x_n \mbox{ is the smallest natural number such that } x_n+x_{n-1} \mbox{ is prime and } x_n\neq x_k \mbox{ for all } k<n.$$ We call $n$ wabbity if $$\{x_1,x_2,\ldots x_n\} = \{1,2,\ldots n\}.$$

Is there an infinitely many wabbity numbers? Is there an effective characterisation of wabbity numbers?

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Bugs Bunny
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    Be vewy vewy quiet... I'm huntin' wabbits... wehehehehe – Asaf Karagila Jun 08 '16 at 19:13
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    Are there any? [yes – $n=2,3,4$ all work] Have you found a few, and then consulted the Online Encyclopedia of Integer Sequences? – Gerry Myerson Jun 08 '16 at 23:16
  • oeis is not responding for me right now. Here's what I get for the sequence of wabbity numbers. http://pastebin.com/wFWTs5eV

    1, 2, 3, 4, 7, 8, 9, 10, 17, 18, 19, 22, 23, 24, 43, 55, 56, 57, 73, 99, 136, 137, 142, 143, 202, 217, 218, 233, 234, 264, 281, 282, 287, 288, 289, 302, 303, 304, 387, 409, 414, 415, 491, 509, 520, 521, 528, 529, 532, 533, 553, 554, 555, 588, 652, 653, 654, 665, 666, 788, 789, 790, 806, 807, 812, 813, 814, 901, 940, 941

     – usul Jun 09 '16 at 00:13
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    OK, the wabbity numbers are here: http://oeis.org/A070942 "Values of n such that the first n terms of http://oeis.org/A055265 constitute a permutation" but little information about them otherwise. – usul Jun 09 '16 at 00:25
  • It seems from oeis that it is unknown if every positive integer eventually appears in A055265, i.e. the sequence ${x_i}$. Of course if some positive integer $m$ does not appear in ${x_i}$, then there can only be finitely many wabbity numbers as all larger than $m$ are disqualified. – usul Jun 09 '16 at 07:07
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    I bet you say that to all the wabbits. Yet it looks like a nice question to prove that all naturals appear in A055265. – Bugs Bunny Jun 09 '16 at 09:27

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