The Wolfram statement: OEIS A104344 ... is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, ... (OEIS A100289). It is known that there can be only a finite number of such primes, but it is not known what the last term is. Going to https://oeis.org/A100289 we find this comment: Write the sum as S(2,n)-1, where S(k,n) = sum_{i=0..n} (i!)^k. Let p=1248829. Because p divides S(2,p-1)-1, p divides S(2,n)-1 for all n >= p-1. Hence there are no primes for n >= p-1.
On Apr 9, 2015, at 3:02 PM, Dan Asimov <asimov@msri.org> wrote:
Also, Wolfram ( http://mathworld.wolfram.com/FactorialSums.html ) states that the sequence
fsq_n := f_n := Sum_{1<=k<=n} (k!)^2
contains only a finite set of primes among its terms. Amazing!
Is there some easy way to see this?