[seqfan] Re: Primes describing digit positions

Éric Angelini bk263401 at skynet.be
Mon Mar 9 23:32:31 CET 2020

Many thanks to Hans -- who just answered to the first
part of my former SeqFan message.
Here is a variant built on the second part of the 
same message -- second part explored by Maximilian
Hasler and declared uninteresting by both me and him
(and the facts).

So, instead of primes describing the positions in S
of prime digits, here is T, a lexico-first seq of
distinct positive non-prime terms describing the
positions of the non-prime digits of T itself:

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

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 herunder.
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):

(follow up with yellow color on my webpage, as usual)

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