1,1

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."

p=17 and q=89 are compatible primes. Intrl(17,89)={1,8,7,9} and a(50)=1879 is a prime interlaced prime.

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 *)

Sequence in context: A238056 A066064 A163759 * A348699 A092622 A129351

Adjacent sequences: A190728 A190729 A190730 * A190732 A190733 A190734

nonn,base

L. Edson Jeffery, May 17 2011

approved