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A375614 Lexicographically earliest infinite sequence of distinct nonnegative pairs of terms that interpenetrate to produce a prime number.

%I A375614 #46 Aug 25 2024 04:28:32
%S A375614 0,11,3,17,2,23,6,13,1,21,4,19,7,27,5,33,8,39,103,10,153,20,107,12,
%T A375614 131,15,109,16,111,26,113,24,101,30,119,14,123,25,117,29,141,22,127,
%U A375614 18,133,31,129,28,121,34,169,36,137,32,167,38,147,35,171,43,157,37,9,41,149,44,159,55,139,46,151,45,163,48,173,42,143,51,187,49,177,52,183,50,161,54,179,47,189
%N A375614 Lexicographically earliest infinite sequence of distinct nonnegative pairs of terms that interpenetrate to produce a prime number.
%C A375614 The term a(n) must always be exactly one digit longer or shorter than the term a(n+1).
%H A375614 Michael S. Branicky, <a href="/A375614/b375614.txt">Table of n, a(n) for n = 1..10000</a>
%H A375614 Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2024/08/combs-and-pharaohs.html">Combs and Pharaohs</a>, personal blog of the author.
%e A375614 Interpenetrate a(1) = 0 and a(2) = 11 to form 101 (a prime number);
%e A375614 interpenetrate a(2) = 11 and a(3) = 3 to form 131 (a prime number);
%e A375614 interpenetrate a(3) = 3 and a(4) = 17 to form 137 (a prime number);
%e A375614 interpenetrate a(4) = 17 and a(5) = 2 to form 127 (a prime number);
%e A375614 interpenetrate a(5) = 2 and a(6) = 23 to form 223 (a prime number);
%e A375614 interpenetrate a(6) = 23 and a(7) = 6 to form 263 (a prime number);
%e A375614 interpenetrate a(7) = 6 and a(8) = 13 to form 163 (a prime number);
%e A375614 interpenetrate a(8) = 13 and a(9) = 1 to form 113 (a prime number);
%e A375614 (...)
%e A375614 interpenetrate a(18) = 39 and a(19) = 103 to form 13093 (a prime number);
%e A375614 (...)
%e A375614 interpenetrate a(167) = 277 and a(168) = 1009 to form 1207079 (a prime number); etc.
%p A375614 Q:= proc(a,b) local La, Lb, i;
%p A375614   La:= convert(a,base,10);
%p A375614   Lb:= convert(b,base,10);
%p A375614   add(La[i]*10^(2*i-2),i=1..nops(La)) + add(Lb[i]*10^(2*i-1),i=1..nops(Lb))
%p A375614 end proc:
%p A375614 f:= proc(n) local d,x;
%p A375614   d:= 1+ilog10(n);
%p A375614   if n::odd then
%p A375614     for x from 10^(d-2) to 10^(d-1) - 1 do
%p A375614       if not(member(x,S)) and isprime(Q(n,x)) then return x fi
%p A375614     od
%p A375614   fi;
%p A375614   for x from 10^d+1 to 10^(d+1) - 1 by 2 do
%p A375614     if not(member(x,S)) and isprime(Q(x,n)) then return x fi
%p A375614   od;
%p A375614 FAIL
%p A375614 end proc:
%p A375614 R:= 0,11: S:= {0,11}: v:= 11:
%p A375614 for i from 2 to 100 do
%p A375614   v:= f(v);
%p A375614   R:= R,v;
%p A375614   S:= S union {v};
%p A375614 od:
%p A375614 R; # _Robert Israel_, Aug 22 2024
%o A375614 (Python)
%o A375614 from sympy import isprime
%o A375614 from itertools import islice
%o A375614 def ip(s, t): return int("".join(x+v for x, v in zip(s, t))+s[-1])
%o A375614 def agen(): # generator of terms
%o A375614     seen, an, found = set(), 0, True
%o A375614     while found:
%o A375614         yield an
%o A375614         seen.add(an)
%o A375614         s = str(an)
%o A375614         d, found = len(s), False
%o A375614         if s[-1] in "1379" and d > 1:
%o A375614             for k in range(10**(d-2), 10**(d-1)):
%o A375614                 if k not in seen and isprime(ip(s, str(k))):
%o A375614                     an, found = k, True
%o A375614                     break
%o A375614         if not found:
%o A375614             for k in range(10**d, 10**(d+1)):
%o A375614                 if k not in seen and isprime(ip(str(k), s)):
%o A375614                     an, found = k, True
%o A375614                     break
%o A375614 print(list(islice(agen(), 90))) # _Michael S. Branicky_, Aug 22 2024
%Y A375614 Cf. A213457, A000040.
%K A375614 base,nonn,look
%O A375614 1,2
%A A375614 _Eric Angelini_ and _Jean-Marc Falcoz_, Aug 22 2024