Multiply Then Add = Primes, Composites

[Edwin Clark]

On Fri, 5 Jan 2007, Leroy Quet wrote:

> Say we have the partition of the first n positive integers into the two 
> sets {a(k)} and {b(k)}.
> 
> We can get the sum S =
> product a(k)'s  + product b(k)'s.
> 
> 1) So, what is the largest prime, if any, S possible for each n?
> 2) What is the least prime, if any, S possible for each n?
> 
> 3) What n' s have no prime S's?
> 
> If we define the product over the empty set to be 1, so that S for n=1 is 
> 2, then I get (by hand) the max primes of:
> 2,3,7,11,43,149,1013

Going out to 20 I get:

  2, 3, 7, 11, 43, 149, 1013, 8069, -infinity, -infinity, 39916801,

        43545611, 566092811, 7925299211, 118879488011, 1609445376013,

        32335220736011, 44771844096143, 582033973248209,

        221172909834240011

Here -infinity indicates there are no prime values for n = 9 and 10

> 
> I get the min primes of:
> 2,3,5,11,23,149,179

and up to 20  here I get:

  2, 3, 5, 11, 23, 149, 179, 1187, infinity, infinity, 3628811,

        43545611, 43545743, 7925299211, 9144576143, 1609445376013,

        32335220736011, 44771844096143, 582033973248209,

        52672757806189


Here infinity indicates there are no prime values for n = 9 and 10.

> 
> I may have very likely made a mistake in calculating these sequences' 
> given terms.

It appears that your values are correct. Please submit the sequences
if you wish.

--Edwin