Review: A067836

[Dean Hickerson]

David Wilson wrote:

> Concerning A067836:
>
> In the description, "nxtprime" and "nxtprm" should be changed to
> "nextprime".

Agreed.

I also agree with the heuristic argument that all terms are prime.  I had
essentially the same thoughts when I edited the sequence in March, which is
why I left the conjectures as they were.

The fact that the terms are distinct (and in fact coprime) was proved by
several people, as described in the sequence's second link.  I'll rewrite
the comment line to clarify that the distinctness has been proved.

> Also, I would like to express some doubt as to the certainty of the
> later elements in the sequence.  f(n) grows very quickly, and is quite
> large by the time we reach n = 58.  I suspect that for larger f(n),
> a(n) = nextprime(f(n)+1) - f(n)) is a probable, not certain, value.  If
> this is so, it should be noted.

Good point.  I've rewritten the Mathematica program using "ProvablePrimeQ",
which is alleged to be reliable.  I've rechecked the given values, and
computed one more.


Here's a question for other Mathematica users.  I'm changing the %t line to:

%t A067836 f[1]=1; f[n_]:=f[n]=f[n-1]a[n-1]; a[1]=2; a[n_]:=a[n]=Module[{i},For[i=2,True,i++,If[ProvablePrimeQ[f[n]+i],Return[i]]]] (* First do <<NumberTheory`PrimeQ` *)

I'd prefer to put the "<<NumberTheory`PrimeQ`" at the beginning of the line
rather than in a comment.  But it doesn't work if it's in the same line as
the first use of ProvablePrimeQ.  E.g. typing

    <<NumberTheory`PrimeQ`; ProvablePrimeQ[7]

gives error messages:

    ProvablePrimeQ::shdw: 
       Symbol ProvablePrimeQ appears in multiple contexts 
        {NumberTheory`PrimeQ`, Global`}; definitions in context 
        NumberTheory`PrimeQ` may shadow or be shadowed by other definitions.

Is there a way to load a package and use one of its functions on the same line?

Dean Hickerson
dean at math.ucdavis.edu