A073641 a(1) = 2; a(n) = smallest prime not included earlier such that concatenation of two successive terms is a prime.

2, 3, 7, 19, 13, 61, 31, 37, 67, 79, 103, 43, 73, 127, 139, 97, 151, 157, 109, 199, 181, 193, 163, 211, 229, 223, 241, 271, 277, 331, 283, 397, 337, 313, 307, 367, 457, 421, 349, 373, 379, 433, 439, 409, 463, 523, 487, 601, 541, 547, 499, 571, 673, 613, 577

Offset: 1

Author

Amarnath Murthy, Aug 09 2002

Keywords

• base
• nonn

Comments

Conjecture: every prime besides 5 is in this list. - Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003

It appears that the terms belong to A007645. There are no primes of form 6k-1 in this sequence. - Alexander Adamchuk, Aug 15 2006

The above conjecture by Cunningham (Apr 11 2003) is false: Since a(2)=3 and a(3)=7 == 1 mod 6, all subsequent terms must also be 1 mod 6 because concatenations of numbers 1 mod 6 with 5 mod 6 are 0 mod 3. - Bob Selcoe, Aug 25 2015

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A075609.

Programs

Maple

N:= 10000: # to get all terms before the first one > N
A[1]:= 2:
Primes:= Vector(select(isprime,[seq(2*i+1 , i=1..floor((N-1)/2))])):
Nprimes:= LinearAlgebra:-Dimension(Primes):
Next:= Vector(Nprimes):
Prev:= Vector(Nprimes):
for i from 1 to Nprimes-1 do Next[i]:= i+1; Prev[i+1]:= i od:
first:= 1:
found:= true:
for n from 2 while found do
  i:= first;
  found:= false;
  while i <> 0 do
    p:= Primes[i];
    if isprime(10^(1+ilog10(p))*A[n-1] + p) then
      found:= true;
      A[n]:= p;
      if i = first then first:= Next[first]
      else Next[Prev[i]]:= Next[i]
      fi;
      if Next[i] <> 0 then
        Prev[Next[i]]:= Prev[i]
      fi;
      break
    fi;
    i:= Next[i];
  od
od:
seq(A[i],i=1..n-2); # Robert Israel, Aug 25 2015

Mathematica

t = {2}; Do[i = 2; While[! PrimeQ[FromDigits[Flatten[IntegerDigits[{Last[t], x = Prime[i]}]]]] || MemberQ[t, x], i++]; AppendTo[t, x], {54}]; t (* Jayanta Basu, Jul 03 2013 *)

Formula

a(n) = A075609(n) for n>1. - Alexander Adamchuk, Aug 15 2006

Extensions

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003