[seqfan] Re: Primes describing digit positions -- follow up

[Éric Angelini]

I wrote (too quickly):

> we must thus wait in using the terms "53" and "67" we've seen above.

... NO! The same argument used to
forbid the digit 2 in T applies for the
digit 5, of course.
Mmmmmh, this leaves very few
possibilities for T to exist.
à+
É.
Catapulté de mon aPhone


> Le 2 mars 2020 à 18:52, Éric Angelini <bk263401 at skynet.be> a écrit :
> 
> Hello SeqFans,
> S = 11, 41, 61, 83, 113, 101, ...
> 11 says: "In position 1, there is 1";
> 41 says: "In position 4, there is 1";
> 61 says: "In position 6, there is 1";
> 83 says: "In position 8, there is 3";
> 113 says: "In position 11, there is 3";
> 101 says: "In position 10, there is 1";
> etc.
> We want the lexico-first seq of distinct
> positive primes "saying" the truth, like
> above, of course.
> 
> But here is the tricky sequence T:
> "Lexico-first seq of distinct positive
> primes describing the position of every 
> prime digit in T".
> 
> What could be a(1) in T?
> 
> Well a(1) cannot start with a digit 1, as
> a(1) = 11 (though a prime term "saying
> the truth") describes the position of 1 
> -- and this 1 is not a prime digit;
> a(1) = 23 describes the position of a prime
> digit, yes, (3) but the digit "2" cannot
> be used in T! Indeed, as 2 is a prime digit,
> any description of it's position in T will
> end in "2" -- leadind to a term that is 
> not prime!
> a(1) = 30 is wrong (not a prime term);
> a(1) = 31 is wrong too as "31" is not the
> description of a prime digit;
> a(1) = 32 is wrong (due to the "2")
> a(1) = 33 --> 36 are not primes;
> a(1) = 37 might be ok -- but as we have
> a "7" in second position, we will write 
> "27" at some point in T -- with a forbidden
> "2"! 
> So, a(1) starts perhaps with "4" and the
> digit "a" (below) cannot be prime;
> 
> T = 4a, . . . 
> 
> This leaves us with a(1) = 41 only (because
> 41 is the only available 2-digit prime); is
> a(2) a 2-digit term? Let's see:
> 
> T = 41, b1, . . (dots are empty positions)
> 
> "b" cannot be 3 (as b is in 3rd position,
> this will lead in the future to "33", which
> is the description of b's position and a
> composite term);
> "b" cannot be 4 (41 is already in T);
> "b" cannot be 5 (51 is not a prime term);
> "b" cannot be 6 (as b is in 3rd position,
> this will lead in the future to "36", which
> is the description of b's position and a
> composite term);
> "b" might be "7" (no apparent contradiction
> yet, it seems):
> 
> T = 41, 71, . . 1 . .
> 
> Is a(3) a 2-digit term? Let's see:
> 
> T = 41, 71, cd, 1 . .
> 
> [this supposes that a(4) is at least a 3-digit
> term, as one can see]; we try "37" for a(3) as 
> this term describes the position of "7" in T:
> 
> T = 41, 71, 37, 1 . .
> 
> This "37" will produce in the future the terms
> 53 and 67 (the description of "37"), which are
> luckily two primes); let's go on filling the
> positions 8 = e and 9 = f:
> 
> T = 41, 71, 37, 1ef, . . .
> 
> What if we use 0 for e and 1 for f? a(4) is a
> prime term at least:
> 
> T = 41, 71, 37, 101, 1 . .
> 
> We see that a(5) is a 3-digit term at least--
> we must thus wait in using the terms "53" and
> "67" we've seen above.
> Etc.
> 
> Best (modulo errors)
> É.
> 
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