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A111011 Primes in A002533.

%I A111011 #25 May 06 2024 02:22:44
%S A111011 7,19,73,241,411379,693110401,80746825394092993,
%T A111011 15848109838244286131940714481,
%U A111011 12238279486576766124458805567902551228138920205718424019,1732765524527243824670663837908764472971413888795440694899,20618141429646301085064054485889973597180353561103310272561,2919234250884982146911220973819117919577845597870261813393281
%N A111011 Primes in A002533.
%C A111011 Starting with the fraction 1/1, generate the sequence of fractions A002533(i)/A002532(i) according to the rule: "add top and bottom to get the new bottom, add top and 6 times bottom to get the new top."
%C A111011 The prime numerators of these fractions are listed here, at locations i= 2, 3, 4, 5, 11, 17, 32, 53,... showing prime(4), prime(8), prime(21), prime(53), prime(34719),..
%C A111011 Is there an infinity of primes in this sequence?
%C A111011 a(17) = A002533(7993), which has 4298 digits so can't be included in a b-file. - _Robert Israel_, May 03 2024
%D A111011 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
%H A111011 Robert Israel, <a href="/A111011/b111011.txt">Table of n, a(n) for n = 1..16</a>
%F A111011 A002533 INTERSECT A000040.
%p A111011 B[0]:= 1: B[1]:= 1: P:= NULL: count:= 0:
%p A111011 for n from 2 while count < 16 do
%p A111011   B[n]:= 2*B[n-1]+5*B[n-2];
%p A111011   if isprime(A[n]) then count:= count+1; P:= P, B[n];  fi
%p A111011 od:
%p A111011 P; # _Robert Israel_, May 03 2024
%t A111011 Select[LinearRecurrence[{2, 5}, {1, 1}, 125] ,PrimeQ[#]&] (* _James C. McMahon_, May 02 2024 *)
%o A111011 (PARI) primenum(n,k,typ) = \\ k=mult,typ=1 num,2 denom. output prime num or denom.
%o A111011 { local(a,b,x,tmp,v); a=1;b=1;
%o A111011 for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,v=a,v=b); if(isprime(v),print1(v","); ) );
%o A111011 print(); print(a/b+.) }
%Y A111011 Cf. A002533, A372491.
%K A111011 easy,nonn
%O A111011 1,1
%A A111011 _Cino Hilliard_, Oct 02 2005
%E A111011 Simplified the definition, listed some A002533 indices. - _R. J. Mathar_, Sep 16 2009
%E A111011 a(10)-a(12) from _James C. McMahon_, May 02 2024