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

[Jonathan Post]

Thank you, Dean and Martin (and thanks to Zak for encouragement).

I've submitted as

A126250

with comments cut & pasted from Dean and Martin's emails as above.

I don't have PARI, and power fails with mean time about a week in this
corner of the grid.  And, without your work, for all we knew there WAS a big
prime in the sequence.  Sometimes a null result (as alleged in
Michaelson-Morley experiment) is important.

Happy Solstice,

Jonathan Vos Post

On 12/21/06, Martin Fuller <martin_n_fuller at btinternet.com> wrote:
>
> 1,2,3,5,6,16,27 and no others.  I think it is not
> worth submitting, because there are no big primes and
> the sequence is fairly easy to calculate (less than a
> week using basic PARI/GP).  But feel free to use the
> result if you can find a use for it.
>
> Martin
>
> --- Jonathan Post <jvospost3 at gmail.com> wrote:
>
> > Thank you, Dean Hickerson. That's why it was nagging
> > at me.  Tony Noe and I
> > had discussed the "n>=762, prime(1)! + ... +
> > prime(n)! is divisible by
> > prime(763) = 5813" fact years ago, when we were both
> > doing things with sums
> > of factorials. I agree that it is somewhat
> > contrived, and I stopped
> > calculating after the sum included 101!, as the
> > increasing factorization
> > times did not look like a good investment on my
> > slow, ancient PC. But, now
> > that you correctly advise that the sequence is
> > finite, I wonder how many
> > solutions there are, and what the largest is. It is
> > clearly much smaller
> > than any of the huge primes being discovered in the
> > past decade. Artificial
> > as it is, at least it is not base!
> >
> > On 12/17/06, Dean Hickerson <dean at math.ucdavis.edu>
> > wrote:
> > >
> > > 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
> > >
> >
>
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20061221/c3df41f1/attachment-0003.htm>