A111010 Primes of the form (3^k - (-1)^k)/4.

2, 7, 61, 547, 398581, 23535794707, 82064241848634269407

Offset: 1

Author

Cino Hilliard, Oct 02 2005

Keywords

• nonn

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)

Crossrefs

Cf. A007658, A015518.

Programs

Mathematica

Do[f=(3^n - (-1)^n)/4; If[PrimeQ[f],Print[{n,f}]],{n,1,577}] (* Alexander Adamchuk, 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+.); }

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)).

a(n) = A015518(A007658(n-1)) for n >= 2. - Amiram Eldar, Jul 04 2024

Extensions

Edited by Alexander Adamchuk, Nov 19 2006