OFFSET
1,1
COMMENTS
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, Nov 19 2006
REFERENCES
John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..14 (terms 1..11 from Alexander Adamchuk, Nov 19 2006)
FORMULA
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)).
MATHEMATICA
Do[f=(3^n - (-1)^n)/4; If[PrimeQ[f], Print[{n, f}]], {n, 1, 577}] (* Alexander Adamchuk, Nov 19 2006 *)
PROG
(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+.); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Oct 02 2005
EXTENSIONS
Edited by Alexander Adamchuk, Nov 19 2006
STATUS
approved