Multiply Then Add = Primes, Composites

[Peter Pein]

Leroy Quet schrieb:
> This is a two-part post, both parts related to the same main idea.
> 
> 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?
> 
...

You might want to add the number of distinct primes you get with this method.
For n=1..20 I get:
1, 1, 2, 1, 3, 1, 3, 2, 0, 0, 3, 1, 4, 1, 2, 1, 1, 1, 1, 3
> 
> 
> ===
> 
> The rest of this message is a post I have made to sci.math.
> Instead of trying to get primes, in this variation we try to get integers 
> with as many divisors as possible.
> 
> In his response, J K Haugland calculated these terms for the max-score 
> sequence defined below:
> 
> 2, 2, 2, 4, 4, 12, 20, 16, 24, 64, 96, 144 
> 
> Anyone able to calculate more terms?
> 
again for n=1..20:
2, 2, 2, 4, 4, 12, 20, 16, 24, 64, 96, 144, 128, 320, 384, 512,
1008, 1296, 1024, 2700

> 
> Thanks,
> Leroy Quet
> 
> 

Peter