A190731 Prime interlaced primes (PIP).

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

Offset

1,1

Comments

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."

Links

Table of n, a(n) for n=1..59.

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] (* Jean-François Alcover, Jun 18 2012 *)

Crossrefs

Sequence in context: A238056 A066064 A163759 * A348699 A092622 A129351

Adjacent sequences: A190728 A190729 A190730 * A190732 A190733 A190734

Keyword

nonn,base

Author

L. Edson Jeffery, May 17 2011

Status

approved