**1**, 1, 4,

**1**, 6,

**1**, 8,

**3**, 1,

**1**,

**3**, 1, 0, 1, ... at positions 1, 2, 3, 4, ...

11 says: "In position 1 is a 1."

41 says: "In position 4 is a 1."

61 says: "In position 6 is a 1."

83 says: "In position 8 is a 3."

113 says: "In position 11 is a 3."

101 says: "In position 10 is a 1."

etc.

Of course, each added prime must be the smallest possible that has not already been used. There's a few early surprises hinting at things to come: 11, 41, 61, 83, 113, 101, 151, 181, 233, 223, 263, 293, 353, 383, 419, 401, 479, 467, 541, 1009, 599, 631, 661, 691, 727, 751, 787, 797, 809, 877, 907, 919, 967, 991, 9001, 1031, ... Term #20 is 1009 because to the end of term #19 we have 53 digits/positions and term #19 says that the next digit (position 54) is a 1. So we need a prime starting with 1 and 1009 is the smallest one that keeps the growing sequence truthful. Term #20 also dictates that in position 100 is a 9. So when we get to term #34 = 991, we now have 99 digits/positions and so the next prime must start with a 9. Why not 997? Because that says that in position 99 is a 7 and we already know that in position 99 is a 1. So we must travel all the way up to 9001 to keep things honest. And that may have repercussions when we get to position 900.

I eventually wrote a Mathematica program that seemed to work extending the sequence. But it was taking a long time finding term #1447. So I had a look at how far it had gotten. Term #1446 was 190901 taking up positions 7006-7011. Perusing the list of prior terms, I saw that positions 7012-7020 and 7022-7024 were already assigned with digits: 191737191?371... Stepping through, 19 is prime, as is 191, but these lie: position 1 is not 9; position 19 is not 1. Continuing, no more primes up to 191737191. Then we can try 1917371911, 1917371913, 1917371917, 1917371919, replacing the ? with 1, 3, 7, 9, but these are not prime either. So we attach the next digit, 3, and replace the ? with 0, 1, 2, 3, ..., 9. We need not go further than 5 because 19173719153, finally, is prime!

So I managed to figure out term #1447 before my program did! In fact, it would

*not*have found it because I had my initial search go up to only 104395301. Here's a graph (click on it) of 1500 terms:

**Updates:**

**Sunday, March 8:**I have run into a second large term at #3868. Term #3867 was 301471 taking up positions 21005-21010. Positions 21011-21020 and 21022-21028 were already assigned with digits: 3713793719?9317373... So #3868 is 371379371929 and #3869 is 31737313.

**Wednesday, March 11:**I have rewritten my original program to run significantly faster. In fact, the new version has already overtaken the number of terms calculated by the old one. Here's an updated graph.

**Monday, March 16:**I've reached 12000 terms and primes strictly greater than 700000.

**Thursday, March 19:**Maximilian Hasler has written PARI/GP code for this sequence which he says computes 10000 terms in a few seconds. Unfortunately, I haven't been able to get it to run.

**Tuesday, March 31:**I've decided to call it quits at 18000 terms. There was another spike at #16966.

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