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%I #10 Jun 18 2012 06:41:37
%S 23,37,53,73,127,131,137,139,151,157,173,179,223,229,233,239,331,337,
%T 421,431,433,457,523,631,677,733,739,751,773,823,829,839,853,859,937,
%U 977,1117,1171,1193,1213,1279,1433,1471,1493,1579,1637,1697,1733,1873,1879,1997,2131,2137,2239,2293,2437,2539,2593,2731
%N Prime interlaced primes (PIP).
%C Let j and k be positive integers with 0<=j-k<=1. Let r and s be positive integers with digit expansions {r_0, r_1, ..., r_j} and {s_0, s_1, ..., s_k}, respectively. The two numbers r, s may be interlaced, denoted by Intrl(r,s), by combining their digit expansions giving Intrl(r,s)={r_0,s_0,...,r_k,s_k}, when j=k, or Intrl(r,s)={r_0,s_0,...,r_k,s_k,r_(k+1)}, when j=k+1. Suppose r and s are prime numbers that can be interlaced. Then a necessary (but not sufficient) condition for the number represented by Intrl(r,s) to be prime is that r and s be congruent modulo 6, in which case r and s are said to be "compatible primes" (for otherwise the sum of their combined digits is divisible by three, when r>3 and s>3). Thus, this sequence is determined by those primes (taken in order, smallest to largest) with digit expansions Intrl(p,q) such that p and q are compatible prime numbers, and a(n) is the n-th "prime interlaced prime."
%e p=17 and q=89 are compatible primes. Intrl(17,89)={1,8,7,9} and a(50)=1879 is a prime interlaced prime.
%t pipQ[p_?PrimeQ] := If[id = IntegerDigits[p]; id[[2]] == 0, False, lid = Length[id]; p1 = FromDigits[ Table[id[[k]], {k, 1, lid, 2}]]; p2 = FromDigits[ Table[id[[k]], {k, 2, lid, 2}]]; PrimeQ[p1] && PrimeQ[p2]]; Select[ Prime /@ Range[9, 400], pipQ] (* _Jean-François Alcover_, Jun 18 2012 *)
%K nonn,base
%O 1,1
%A _L. Edson Jeffery_, May 17 2011
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