(2! + 3! + 5! + ... + 53!) = prime * 2^4

[Dean Hickerson]

Jonathan Post wrote:

> Numbers n such that the sum of the first n factorials
> of primes is a power of 2, or a prime times a power of 2.
...
> a(n) = 1, 2, 3, 5, 6, 16, ...
>
> I looked at sums up through 101!

I hesitate to comment, since this seems like another artifical sequence
to me.  However:

First note that from 2!+3!+5!+7! on, the power of 2 will always be 2^4.
So after the first 3 terms, an equivalent definition is that the sum
has the form 16p for some prime p.  Also, the sequence is finite:  For all
n>=762, prime(1)! + ... + prime(n)! is divisible by prime(763) = 5813,
and is therefore not of the form 16p.

Dean Hickerson
dean at math.ucdavis.edu