F26 in UPINT

[David Wilson]

I appreciate the appreciation, but I do not merit a mention in UPINT. I 
merely re-asked a question that had clearly been considered years earlier. 
All that had changed was the power of the machines that could be brought to 
bear against the problem. Anyone who had revisited the problem would 
certainly have inspired the same results, it is those who do the work that 
deserve the credit.

Just for summary's sake, is the following correct?

    a(n) = A005245(n)
    smin(n) = MIN{a(x)+a(y): x+y = n, x <= y < n}if n >= 2.
    pmin(n) = MIN{a(x)+a(y): xy = n, x <= y < n} if n composite.

giving us the recursive definition

    a(n) = 1
        1; if n = 1
        smin(n); if n prime
        min(smin(n), pmin(n)); if n composite

and the recently discovered numbers would be (?)

46 = smallest example of smin(n) < pmin(n)
21080618 = smallest example of smin(n) < 1+a(n-1)
353942783 = smallest example of n prime, smin(n) < 1+a(n-1) <=> a(n) < 
1+a(n-1)
516743639 = smallest example of smin(n) < min(1+a(n-1), pmin(n))

----- Original Message ----- 
From: "Max Alekseyev" <maxale at gmail.com>
To: "Richard Guy" <rkg at cpsc.ucalgary.ca>
Cc: "Martin Fuller" <martin_n_fuller at btinternet.com>; "David Wilson" 
<davidwwilson at comcast.net>
Sent: Friday, February 01, 2008 3:51 PM
Subject: Re: F26 in UPINT


> As of attribution, I think it should go to David for his remarkably
> inspiring conjectures (despite that they turned out to be incorrect),
> to CD_Eater (I can ask him for his credentials if you decide to
> mention him in UPINT) for his initial breakthrough and counterexample
> to David's first conjecture, and Martin Fuller who finally disproved
> David's second conjecture and the conjecture in F26.
>
> Regards,
> Max