A111011 Primes in A002533.

7, 19, 73, 241, 411379, 693110401, 80746825394092993, 15848109838244286131940714481, 12238279486576766124458805567902551228138920205718424019, 1732765524527243824670663837908764472971413888795440694899, 20618141429646301085064054485889973597180353561103310272561, 2919234250884982146911220973819117919577845597870261813393281

Offset: 1

Author

Cino Hilliard, Oct 02 2005

Keywords

• easy
• nonn

Comments

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."

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),..

Is there an infinity of primes in this sequence?

a(17) = A002533(7993), which has 4298 digits so can't be included in a b-file. - Robert Israel, May 03 2024

References

• John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Links

• Robert Israel, Table of n, a(n) for n = 1..16

Crossrefs

Cf. A002533, A372491.

Programs

Maple

B[0]:= 1: B[1]:= 1: P:= NULL: count:= 0:
for n from 2 while count < 16 do
  B[n]:= 2*B[n-1]+5*B[n-2];
  if isprime(A[n]) then count:= count+1; P:= P, B[n];  fi
od:
P; # Robert Israel, May 03 2024

Mathematica

Select[LinearRecurrence[{2, 5}, {1, 1}, 125] ,PrimeQ[#]&] (* James C. McMahon, May 02 2024 *)

PARI

primenum(n,k,typ) = \\ k=mult,typ=1 num,2 denom. output prime num or denom.
{ local(a,b,x,tmp,v); a=1;b=1;
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","); ) );
print(); print(a/b+.) }

Formula

A002533 INTERSECT A000040.

Extensions

Simplified the definition, listed some A002533 indices. - R. J. Mathar, Sep 16 2009

a(10)-a(12) from James C. McMahon, May 02 2024