# [seqfan] Doubly self-describing sequence: a(n) = odd integer and odd digit position in S

Eric Angelini Eric.Angelini at kntv.be
Mon Oct 28 18:19:14 CET 2013

```Hello SeqFans,
Why not "merge" those two constraints and build and new seq obeying
both of them:

http://oeis.org/A079313
a(n) is taken to be the smallest positive integer not already present
which is consistent with the condition:
"n is a member of the sequence if and only if a(n) is odd".

http://oeis.org/A125132
Self-describing sequence: sequence starts with a(1) = 1 and a(n) is chosen
to be the smallest positive number not already in the sequence such that
the assertion:
"sequence gives the positions of the odd digits when the sequence is read
as a string of digits"
is true.

We would thus build S in which every a(n) would "say":
- the a(n)th integer of S is odd
- the a(n)th digit of S is odd
and where a(n) is chosen to be the smallest positive number not already in

I guess S starts pretty much like A125132 -- then diverges after a(n)=99:

A125132:
1, 3, 5, 2, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 110, 10, 12, 14, 16, 111, 28, 30, 32, 34, 36, 48, 50, 52,...

S:
1, 3, 5, 2, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 111, 10, 101, 12, 103, 14, 105, 16, 113, 28,...

S is easy to build (by computer) and S is surely infinite (I can't prove it
though).

The sequence T could achieve the same task with the "even integers/digits"
double condition.
Best,
É.

```