

A190731


Prime interlaced primes (PIP).


0



23, 37, 53, 73, 127, 131, 137, 139, 151, 157, 173, 179, 223, 229, 233, 239, 331, 337, 421, 431, 433, 457, 523, 631, 677, 733, 739, 751, 773, 823, 829, 839, 853, 859, 937, 977, 1117, 1171, 1193, 1213, 1279, 1433, 1471, 1493, 1579, 1637, 1697, 1733, 1873, 1879, 1997, 2131, 2137, 2239, 2293, 2437, 2539, 2593, 2731
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OFFSET

1,1


COMMENTS

Let j and k be positive integers with 0<=jk<=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 nth "prime interlaced prime."


LINKS



EXAMPLE

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


MATHEMATICA

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] (* JeanFrançois Alcover, Jun 18 2012 *)


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



