Non-primes describing non-prime digits' positions in T

Non-primes describing non-prime digits' positions in T

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, 270, ...

Recap for the absents:

“20” must be read: “At position 2, there is a 0”. And indeed, there is, when considering T as a string of concatenated digits.

“40” reads: “At position 4, there is a 0” – which is true.

“34” reads: “At position 3, there is a 4” – which is also true.

…

“200” reads: “At position 20, there is a 0” – which is true (the central 0 in 300); etc.

We see with the 200 example that the last digit “d” is what we describe and the string before “d” is the position of “d” in T (T stands for tricky).

The other rules have not changed: we describe only non-prime digits; we use in T only non-prime terms; T is the lexicographically earliest seq of distinct positive integers with this property.

Is this seq really tricky? Oh yeaaah! See hereunder.

Why is a(1) not equal to 1?

Because no term of T can have less than 2 digits.

Why is a(1) not equal to 10?

Because there is no 0 in position 1.

Why is a(1) not equal to 11?

Because 11 is a prime – which is forbidden.

Why is a(2) not equal to 12?

Because we don’t describe prime digits’ positions – and 2 is such a pariah.

Why is a(3) not equal to 33?

Because we don’t describe prime digits’ positions – and 3 is another such pariah (like 5 and 7).

Why is a(3) not equal to 64?

Because we always extend T with the smallest available term not leading to a contradiction – and 34 is such a term.

Why is a(7) = 200? Is it impossible to find a smaller term?

Because yes, it is impossible – look. We yellow the 6 digits that are described by the terms a(1) to a(6):

T = 20, 40, 34, 64, 76, 84, ...

As we don’t describe prime digits, we will skip the “7” of “76”, focusing on the 6. This 6 is in position 10. So, we must at some point include 106 in T. Is it possible right now? Let’s see:

T = 20, 40, 34, 64, 76, 84, 106,...

No, this is wrong – because we will be obliged later (by definition) to describe the position of the 1, of the 0 and of the 6 (of “106”), as they are non-prime digits.

But… the 1 is in position 13 – which means that we will have, at some point in the future again, to include the term 131 in T – and as 131 is prime, this is forbidden. This means that we cannot have a “1” in position 13. Thus, we shall have at least a “2” starting a(7). This leads to a(7) = 200 (as the zeros of 200 won’t cause any problem in the future description of their position). And, in general, we will enjoy the same "description comfort" with any other even digit – plus the digit 5: such digits, preceded by any integer/describer, will never be primes: the concatenations [k0], [k2], [k4], [k5], [k6] and [k8] are composites for any k.

  

Ok, we go on with T. We notice that a(7) = 200 (starting with a "2" instead of a forbidden "1") seems to fit. It says: “At position 20, there is a zero”. We must immediately put a 0 in position 20 (we will fill the “empty spaces” between 200 and this 0 in the best lexico-manner later):

T = 20, 40, 34, 64, 76, 84, 200, . . . . 0 . . . (dots are empty spaces)

Well, what about the first yet non-described non-prime digit of T, namely the 6 of “76”? His “describer” is 106 – can we extend T with it and have a(8) = 106? Yes, no future contradiction with the "1" of 106 as this "1", in position 16 will generate 161, a composite number:

T = 20, 40, 34, 64, 76, 84, 200, 106, . 0 . . .

Now, the only extension of T, at this point, is with a(9) = 300, due to the 0 in position 20; this 300 installs a (yellow) zero in position 30 in T:

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, . . . . . . . . 0 . . . 

The next digit we would like to describe now is the "8" of 84, above – in position 11. Can we extend T with 118? Let's see:

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, . . . . . 0 . . . 

We must check if the 1's of 118 will produce future primes or not: as 221 and 231 are not primes, we can indeed extend T with 118 – good! What comes next? The description of the digit "4" (in the same 84 above); this "4" is in position 12 – thus we will try to use the term 124 for a(11):

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 124, . . 0 . . . 

We check immediately if 124 will generate a prime in the future – and indeed it will, as 251 is prime. So, no 124 now:

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 124, . . 0 . . . 

What about describing instead the position of the "2" in 200, above? No, remember that "2" is a prime digit– and we don't describe them. What about the first "0" of 200? This would extend T with 140 instead of  124 – but wait, as no term of T can have a "1" in position 25, we must at least start a(11) with a "2". Which leads to 210, the description of the second "0" of 300:

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, . . 0 . . .

Does it fit? Will the "1" in the middle of 210 generate a future prime? No, as 261 is a composite – we thus keep 210 where it is, as a(11):

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, . . 0 . . .

The next term of T ends in "0" (forced); it will be 270 (describing the "0" of 210) – as the "7" in the middle of this term will generate 267 in the future, which is not a prime. We yellow the "0" of 210:

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, 270, . . .

Ok, at this point we see that the description of "4" (in 84, above) is still hanging. Carole Dubois will tell us soon if 124 (the hanging description) will extend T now ... or later!

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, 270, 124 ?
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Update, March 12, 2020

Carole was quick to answer: "No, 124 is not welcomed now! See the first 100 terms of the sequence (soon in the OEIS):

T = 20, 40, 34, 64, 76, 84, 200, 106, 300, 118, 210, 270, 248, 124, 140, 324, 221, 231, 261, 338, 150, 170, 186, 364, 384, 341, 390, 371, 424, 451, 161, 481, 506, 511, 548, 570, 600, 551, 581, 611, 628, 636, 656, 664, 688, 714, 694, 749, 721, 750, 794, 781, 814, 824, 841, 866, 851, 871, 884, 898, 1020, 920, 901, 936, 1036, 951, 961, 984, 2004, 1040, 1050, 998, 2020, 1081, 1206, 2030, 1108, 1236, 1216, 2044, 1111, 2060, 1126, 1131, 1156, 1178, 1246, 1186, 2051, 1256, 1141, 1264, 1276, 2074, 2080, 1288, 2091, 1298, 1324, 1336, ...