A073641 - OEIS

%I #29 Aug 26 2015 09:02:45

%S 2,3,7,19,13,61,31,37,67,79,103,43,73,127,139,97,151,157,109,199,181,

%T 193,163,211,229,223,241,271,277,331,283,397,337,313,307,367,457,421,

%U 349,373,379,433,439,409,463,523,487,601,541,547,499,571,673,613,577

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

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

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

%C 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

%H Robert Israel, <a href="/A073641/b073641.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A075609(n) for n>1. - _Alexander Adamchuk_, Aug 15 2006

%p N:= 10000: # to get all terms before the first one > N

%p A[1]:= 2:

%p Primes:= Vector(select(isprime,[seq(2*i+1 , i=1..floor((N-1)/2))])):

%p Nprimes:= LinearAlgebra:-Dimension(Primes):

%p Next:= Vector(Nprimes):

%p Prev:= Vector(Nprimes):

%p for i from 1 to Nprimes-1 do Next[i]:= i+1; Prev[i+1]:= i od:

%p first:= 1:

%p found:= true:

%p for n from 2 while found do

%p   i:= first;

%p   found:= false;

%p   while i <> 0 do

%p     p:= Primes[i];

%p     if isprime(10^(1+ilog10(p))*A[n-1] + p) then

%p       found:= true;

%p       A[n]:= p;

%p       if i = first then first:= Next[first]

%p       else Next[Prev[i]]:= Next[i]

%p       fi;

%p       if Next[i] <> 0 then

%p         Prev[Next[i]]:= Prev[i]

%p       fi;

%p       break

%p     fi;

%p     i:= Next[i];

%p   od

%p od:

%p seq(A[i],i=1..n-2); # _Robert Israel_, Aug 25 2015

%t 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 *)

%Y Cf. A000040, A075609.

%K base,nonn

%O 1,1

%A _Amarnath Murthy_, Aug 09 2002

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

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