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A383743 a(n) is the smallest prime not yet in the sequence that satisfies the following: for some pair of different digits i and j in a(n-1), i preceding j (from left to right), j precedes i in a(n). Leading 0s are not allowed; a(1)=13. See Comments for details.

%I A383743 #21 Jun 15 2025 22:53:54
%S A383743 13,31,103,101,107,71,17,271,127,211,1021,109,191,19,491,139,131,113,
%T A383743 311,137,73,37,173,307,373,317,163,61,167,461,149,41,1049,241,421,
%U A383743 1123,251,151,157,521,257,523,353,53,359,193,239,293,349,43,347,431
%N A383743 a(n) is the smallest prime not yet in the sequence that satisfies the following: for some pair of different digits i and j in a(n-1), i preceding j (from left to right), j precedes i in a(n). Leading 0s are not allowed; a(1)=13. See Comments for details.
%C A383743 To obtain a(n), the digits i and j in a(n-1) are chosen among all possible pairs of different digits in a(n-1) so that they produce the smallest prime not yet in the sequence according to the precedence requirement in the definition.  Naturally, to obtain a(n+1), the process is repeated but now a different pair of digits k, l in a(n) may be chosen to produce a(n+1).
%C A383743 Note that it is enough that there exists a pair of digits i and j in a(n) that satisfy the precedence requirement; not every pair of such digits must satisfy the requirement in the case of multiple occurrences of such digits in a(n) (see examples).
%H A383743 Robert Israel, <a href="/A383743/b383743.txt">Table of n, a(n) for n = 1..10000</a>
%H A383743 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a383/A383743.java">Java program</a> (github)
%e A383743 In a(2)=31 3 precedes 1, hence a(3)=103 since it is the smallest prime not yet in the sequence in which the digit 1 precedes 3.
%e A383743 In a(3)=103 1 precedes 0, hence a(4)=101 since it is the smallest prime not yet in the sequence in which the digit 0 precedes 1 (last two digits).
%e A383743 In a(10)=211 2 precedes 1, hence a(11)=1021 since it is the smallest prime not yet in the sequence in which the digit 1 precedes 2 (first and third digits).
%e A383743 In a(11)=1021 0 precedes 1 (second and fourth digits), hence a(12)=109 since it is the smallest prime not yet in the sequence in which the digit 1 precedes 0 (first and second digits).
%p A383743 P:= select(isprime,[seq(i,i=13 .. 1000,2)]):
%p A383743 g:= proc(n) local L, R,i,j;
%p A383743    L:= convert(n,base,10);
%p A383743    R:= select(t -> t[2] <> t[1], {seq(seq([L[i],L[j]],j=1..i-1),i=1..nops(L))});
%p A383743 end proc:
%p A383743 R:= 13: r:= 13: Cands:= subsop(1=NULL,P):
%p A383743 do qr:= map(t -> [t[2],t[1]],g(r)): found:= false;
%p A383743    for i from 1 to nops(Cands) do
%p A383743      if g(Cands[i]) intersect qr <> {} then
%p A383743        r:= Cands[i]; R:= R,r; Cands:= subsop(i=NULL,Cands); found:= true; break
%p A383743      fi
%p A383743    od;
%p A383743 until not found:
%p A383743 R; # _Robert Israel_, Jun 09 2025
%Y A383743 Cf. A107801, A381130.
%K A383743 nonn,base
%O A383743 1,1
%A A383743 _Enrique Navarrete_, May 08 2025