A066065 - OEIS

%I #16 Dec 28 2017 03:37:02

%S 3,7,3,3,3,7,3,3,3,3,3,3,11,3,23,23,3,3,3,29,3,7,11,23,7,3,3,11,3,11,

%T 7,47,3,13,3,31,31,7,29,3,11,19,3,3,3,3,3,7,3,3,3,3,7,11,17,3,3,3,7,

%U 11,3,11,13,23,7,23,3,3,29,13,3,3,3,3,3,3,17,19,3,3,11,7,17,7,7,71,3,37,41

%N a(n) = smallest prime q such that in decimal notation the concatenation prime(n)q yields a prime ( = A066064(n)).

%C Conjecture: a(k) < prime(k) for k > 2.

%C a(n)=3 if and only if prime(n) is in A023238. - _Robert Israel_, Dec 27 2017

%H Robert Israel, <a href="/A066065/b066065.txt">Table of n, a(n) for n = 1..20000</a> (n=1..1000 from Harry J. Smith)

%e A000040(13) = 41; for the first four primes 2, 3, 5 and 7 we get 412, 413, 415 and 417, which are all composite, but with the 5th prime we have 4111 = A066064(13), so a(13) = 11.

%p N:= 100: # to get a(1)..a(N)

%p P:= Vector(N,ithprime):

%p A:= Vector(N):

%p q:= 2:

%p Agenda:= {$1..N}:

%p while Agenda <> {} do

%p   q:= nextprime(q);

%p   m:= 10^(ilog10(q)+1);

%p   L,Agenda:= selectremove(t -> isprime(P[t]*m+q), Agenda);

%p   A[convert(L,list)]:= q;

%p od:

%p convert(A,list); # _Robert Israel_, Dec 27 2017

%o (PARI) digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d-1) > x, d--)); return(d) }

%o Concat(a, b)= { return(a*10^digitsIn(b) + b) }

%o { for (n = 1, 1000, p=prime(n); q=2; while(!isprime(Concat(p, q)), q=nextprime(q + 1)); write("b066065.txt", n, " ", q) ) } \\ _Harry J. Smith_, Nov 09 2009

%Y Cf. A000040, A023238, A066064.

%K base,nonn

%O 1,1

%A _Reinhard Zumkeller_, Dec 01 2001