%N Integers that concatenate 3 counts: the number of terms in the sequence so far, the number of primes in the sequence so far, the number of digits in the sequence so far, with a(1)= 113. The sequence is always extended with the smallest available integer not leading to a contradiction or a dead end.
%C What would a dead end be? We have here a(8) = 8329 though a(8) = 8229 is sound (and lexicographically better); but 8229 brings the sequence to an end after a(9) = 9233 and a(10) = 10238 (indeed, 10238 can't be followed by anything without a contradiction). It is unknown by the authors if this sequence is finite or not.
%H Carole Dubois, <a href="/A309617/b309617.txt">Table of n, a(n) for n = 1..1002</a>
%H Carole Dubois, <a href="/A309617/a309617.jpg">graph with axe Y logarithmic</a>
%e a(1) = 113 must be read like this: "so far in the sequence there are 1 integer, 1 prime and 3 digits", which is true;
%e a(2) = 216 must be read like this: "so far in the sequence there are 2 integers, 1 prime and 6 digits", which is true;
%e a(3) = 319 must be read like this: "so far in the sequence there are 3 integers, 1 prime and 9 digits", which is true;
%e a(6) = 6221 must be read like this: "so far in the sequence there are 6 integers, 2 primes and 21 digits", which is true (the 2 primes are 113 and 6221 itself);
%A _Eric Angelini_ and _Carole Dubois_, Aug 10 2019