# [seqfan] Primes describing digit positions

Éric Angelini bk263401 at skynet.be
Mon Mar 2 18:52:21 CET 2020

```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?

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.
```