|
|
A111010
|
|
Primes of the form (3^k - (-1)^k)/4.
|
|
3
|
|
|
|
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
|
Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.
|
|
LINKS
|
|
|
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
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|