1,1

The next term is too large to include.

Is there an infinity of primes in this sequence?

All a(n), except a(1) = 2, are primes of the form (3^k + 1)/4. Corresponding numbers k such that (3^k + 1)/4 is prime are listed in A007658(n) = {3, 5, 7, 13, 23, 43, 281, 359, 487, 577, ...}. All such numbers k are primes. a(1) = 2 is the only prime of the form (3^k - 1)/4. - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006

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

Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006, Table of n, a(n) for n = 1..11

Given a(0)=1, b(0)=1 then for i=1, 2, .. a(i)/b(i) = (a(i-1)+2*b(i-1)) /(a(i-1) + b(i-1)).

Do[f=(3^n - (-1)^n)/4; If[PrimeQ[f], Print[{n, f}]], {n, 1, 577}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006

(PARI) primenum(n, k, typ) = \ k=mult, typ=1 num, 2 denom. ouyput 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+.) }

Cf. A007658, A015518.

Adjacent sequences: A111007 A111008 A111009 this_sequence A111011 A111012 A111013

Sequence in context: A000892 A065397 A046846 this_sequence A089307 A102896 A088107

easy,nonn

Cino Hilliard (hillcino368(AT)gmail.com), Oct 02 2005

Edited by Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 19 2006