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

%I A073641 #29 Aug 26 2015 09:02:45
%S A073641 2,3,7,19,13,61,31,37,67,79,103,43,73,127,139,97,151,157,109,199,181,
%T A073641 193,163,211,229,223,241,271,277,331,283,397,337,313,307,367,457,421,
%U A073641 349,373,379,433,439,409,463,523,487,601,541,547,499,571,673,613,577
%N A073641 a(1) = 2; a(n) = smallest prime not included earlier such that concatenation of two successive terms is a prime.
%C A073641 Conjecture: every prime besides 5 is in this list. - Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003
%C A073641 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 A073641 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 A073641 Robert Israel, <a href="/A073641/b073641.txt">Table of n, a(n) for n = 1..10000</a>
%F A073641 a(n) = A075609(n) for n>1. - _Alexander Adamchuk_, Aug 15 2006
%p A073641 N:= 10000: # to get all terms before the first one > N
%p A073641 A[1]:= 2:
%p A073641 Primes:= Vector(select(isprime,[seq(2*i+1 , i=1..floor((N-1)/2))])):
%p A073641 Nprimes:= LinearAlgebra:-Dimension(Primes):
%p A073641 Next:= Vector(Nprimes):
%p A073641 Prev:= Vector(Nprimes):
%p A073641 for i from 1 to Nprimes-1 do Next[i]:= i+1; Prev[i+1]:= i od:
%p A073641 first:= 1:
%p A073641 found:= true:
%p A073641 for n from 2 while found do
%p A073641   i:= first;
%p A073641   found:= false;
%p A073641   while i <> 0 do
%p A073641     p:= Primes[i];
%p A073641     if isprime(10^(1+ilog10(p))*A[n-1] + p) then
%p A073641       found:= true;
%p A073641       A[n]:= p;
%p A073641       if i = first then first:= Next[first]
%p A073641       else Next[Prev[i]]:= Next[i]
%p A073641       fi;
%p A073641       if Next[i] <> 0 then
%p A073641         Prev[Next[i]]:= Prev[i]
%p A073641       fi;
%p A073641       break
%p A073641     fi;
%p A073641     i:= Next[i];
%p A073641   od
%p A073641 od:
%p A073641 seq(A[i],i=1..n-2); # _Robert Israel_, Aug 25 2015
%t A073641 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 A073641 Cf. A000040, A075609.
%K A073641 base,nonn
%O A073641 1,1
%A A073641 _Amarnath Murthy_, Aug 09 2002
%E A073641 More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 11 2003
%E A073641 Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003