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A066065 a(n) = smallest prime q such that in decimal notation the concatenation prime(n)q yields a prime ( = A066064(n)).

%I A066065 #19 Dec 08 2024 22:10:26
%S A066065 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 A066065 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 A066065 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 A066065 a(n) = smallest prime q such that in decimal notation the concatenation prime(n)q yields a prime ( = A066064(n)).
%C A066065 Conjecture: a(k) < prime(k) for k > 2.
%C A066065 a(n)=3 if and only if prime(n) is in A023238. - _Robert Israel_, Dec 27 2017
%H A066065 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 A066065 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 A066065 N:= 100: # to get a(1)..a(N)
%p A066065 P:= Vector(N,ithprime):
%p A066065 A:= Vector(N):
%p A066065 q:= 2:
%p A066065 Agenda:= {$1..N}:
%p A066065 while Agenda <> {} do
%p A066065   q:= nextprime(q);
%p A066065   m:= 10^(ilog10(q)+1);
%p A066065   L,Agenda:= selectremove(t -> isprime(P[t]*m+q), Agenda);
%p A066065   A[convert(L,list)]:= q;
%p A066065 od:
%p A066065 convert(A,list); # _Robert Israel_, Dec 27 2017
%o A066065 (PARI) a(n) = { my(p=prime(n)); forprime(q=3, oo, if(isprime(p*10^(logint(q,10)+1) + q), return(q))) } \\ _Harry J. Smith_, Nov 09 2009
%Y A066065 Cf. A000040, A023238, A066064.
%K A066065 base,nonn
%O A066065 1,1
%A A066065 _Reinhard Zumkeller_, Dec 01 2001