a(n) = smallest prime >a(n-1) such that sum of first n terms is Fib no.
Jack Brennen
jack at brennen.net
Sat Dec 21 01:34:17 CET 2002
Neil Fernandez wrote:
>
> 2,3,29,199,2351,24155233, next term >10^15
>
> a(n) = smallest prime > a(n-1) such that sum of first n terms is a
> Fibonacci number
>
> a(7), if it exists, > 10^15
I'll give the Fibonacci indices, because it's a more compact
representation of the sequence... I'm pretty sure it goes like:
ai(n) = 0,3,5,9,13,18,37,384,569,2760...
(Only "pretty sure" because I haven't rigorously proved the
numbers primes...)
This sequence is defined:
ai(0) = 0
ai(1) = 3
ai(n+1) = the smallest x such that U(x)-U(ai(n)) is both prime
and greater than U(ai(n))-U(ai(n-1))
Where U(x) is the Lucas sequence defined by (P,Q)=(1,-1) --
the Fibonacci numbers.
The sequence a(n) in the quoted post can be defined for n>=1 as:
a(n) = U(ai(n))-U(ai(n-1))
a(7), a(8), and a(9) have 80, 119, and 577 digits respectively.
Jack
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