# [seqfan] Re: Jump and land on a prime

Benoît Jubin benoit.jubin at gmail.com
Thu Dec 22 23:50:49 CET 2011

```This made me think of these two sequences:

* Lexicographically smallest injective sequence of positive integers
such that a(n+a(n)) is prime for all n.
1,2,3,5,4,7,6,8,11,9,10,12,13

* Lexicographically smallest injective sequence of positive integers
such that a(n) is prime if and only if there is a k such that
n=k+a(k).
1,2,4,3,6,8,5,9,10,12,7,11,14,13
This is the same if we only ask "a(n) is prime only if there is a k
such that n=k+a(k)".

* In both sequences, both primes and composites appear in increasing
order.  These sequences are permutations of the positive integers.
[note that these sequences correspond to "move to the right by m
steps", which seems more natural than "jump over p terms" (obviously,
p=m-1)]

Benoit

On Thu, Dec 22, 2011 at 9:30 AM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> Hello SeqFans,
>
> S=1,2,3,4,5,6,7,8,11,9,13,12,17,10,19,15,23,14,16,29,31,18,20,21,37,22,24,25,26,27,41,43,47,28,53,59,33,30,35,32,61,34,39,67,42,71,44,36,73,79,45,83,89,97,...
>
> Pick any S(n) and jump over S(n) terms: you'll land on a Prime.
> You can jump left or right, it'll do.
> Examples:
> "1" leads you to "3" (you have jumped over 1 term)
> "6" leads you to "17" (you have jumped over 6 terms)
> "10" leads you to "3" or "37" (you have jumped over 10 terms, left or right).
> S is a permutation of the natural numbers - and the primes appear in their natural order.
> Not sure of the last terms, though.
> Best,
> É.
>
>
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```