[seqfan] 3-label integers

[Éric Angelini]

Hello SeqFans,
here is a completely artificial and useless seq S: 

S = 113, 216, 319, 4113, 5117, 6221, 7225, 8329, 9333, 10338, 11343, 12348, 13353, 14358, 15363, 16368, 17373, 18378, 19383, 20388, 21393, 22398, 233104, 243110, 253116, 263122, 273128, 283134, 293140, 303146, 313152, 323158, 333164, 343170, 353176, 363182, 373188, 383194, 393200, 403206, 413212, 423218, 433224, 443230, 453236, 463242, 473248, 483254, 493260, 503266, 513272, 523278, 533284, 543290, 553296, 563302, 573308, 583314, 593320, 603326, 613332, 623338, 633344, 643350, 653356, 663362, 673368, 683374, 693380, 703386, 713392, 723398, 733404, 743410, 753416, 763422, 773428, 783434, 793440, 803446, 813452, 823458, 833464, 843470, 853476, 863482, 873488, 883494, 893500, 903506, 913512, 923518, 933524, 943530, 953536, 963542, 973548, 983554, 993560, 1004567, 1014574, 1024581, 1034588, 1044595, 1054602, 1064609, 1074616, 1084623, 1094630, 1104637, 1114644, 1124651, 1134658, 1144665, 1154672, 1164679, 1174686, 1184693, 1194700, 1204707, 1214714, 1224721, 1234728, 1244735, 1254742, 1264749, 1274756, 1285763, 1295770, 1305777, 1315784, 1326791, 1336798, 1346805, 1356812, 1366819, 1376826, 1386833, 1396840, 1406847, 1416854, 1426861, 1436868, 1446875, 1456882, 1467889, 1477896, 1487903, 1497910, 1507917,...

S was computed by my friend Carole Dubois to which I've 
sent yesterday the hereunder artificial and useless idea:

> Take any integer L (like 1507917, the last one here):
  L is the concatenation of [the index n of L]+[the number
  of primes in S so far]+[the number of digits in S so far].

For L = 1507917 we thus have: 
    n = 150  
           7 primes so far in S
            917 digits so far in S.

The fun (?) part is that sometimes, in trying to extend S,
two distinct integers are possible -- opening two possible
"branches" to S.
If the lexico-first branch stops, you can then switch to
the other one -- hoping that S doesn't stop too quickly.

How far could this sequence be extended, before it definitely
stops?
Best,
É.