# [seqfan] Re: Sequence A037153

Richard Guy rkg at cpsc.ucalgary.ca
Thu Mar 4 21:12:10 CET 2010

Here's a quote from A2 in UPINT:

Fortune's conjecture

Let $q$ be the least prime greater than $p\#$.
Then Reo F.~Fortune conjectured that $q-p\#$ is
prime (or 1) for all primes $p$. It is clear that it
can only be divisible by primes greater than $p$, and
Selfridge observes that
the truth of the conjecture would follow from
Schinzel's formulation of Cramer's conjecture,
(p.\,7 of his 1961 paper quoted at the head of this
chapter) that for $x\geq8$ there is always a prime
between $x$ and $x+(\ln x)^2$. Stan Wagon has
calculated the first 100 fortunate primes:

\begin{tabular}
3 & 5 & 7 & 13 & 23 & 17 & 19 & 23 & 37 & 61 & 67 & 61 & 71 &
47 & 107 & 59 & 61 & 109 & 89 & 103 \\
79 & 151 & 197 & 101 & 103 & 233 & 223 & 127 & 223 & 191 & 163 &
229 & 643 & 239 & 157 & 167 & 439 & 239 & 199 & 191 \\
199 & 383 & 233 & 751 & 313 & 773 & 607 & 313 & 383 & 293 & 443 &
331 & 283 & 277 & 271 & 401 & 307 & 331 & 379 & 491 \\
331 & 311 & 397 & 331 & 353 & 419 & 421 & 883 & 547 & 1381 & 457 &
457 & 373 & 421 & 409 & 1061 & 523 & 499 & 619 & 727 \\
457 & 509 & 439 & 911 & 461 & 823 & 613 & 617 & 1021 & 523 & 941 &
653 & 601 & 877 & 607 & 631 & 733 & 757 & 877 & 641
\end{tabular}

under the assumption that the very large probable primes involved are
genuine primes. The answers to the questions are probably yes'',
but it does not seem conceivable that such conjectures will come
within reach either of computers or of analytical tools in the
foreseeable future.  Schinzel's conjecture has been attributed
to Cram\'er, but Cram\'er conjectured (see ref.\ at {\bf A8})
$$\hbox{?}\qquad\limsup_{n\rightarrow\infty}{p_{n+1}-p_n\over (\ln p_n)^2}=1\qquad ?$$
Schinzel notes that this doesn't imply the
existence of a prime between $x$ and $x+(\ln x)^2$, even for
sufficiently large $x$.

R.

On Thu, 4 Mar 2010, N. J. A. Sloane wrote:

> Dear Seq Fans,  the definition of A037153 is
>
> %N A037153 a(n)=p-n!, where p is the smallest prime > n!+1.
> That is, let p = smallest prime > n!+1, then a(n) = p - n!.
>
> There is a comment saying:
>
> %C A037153 Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 541 terms are primes. Are all terms prime?
>
> So there is a potentially different sequence, call it S, defined by:
>
> Let p = smallest prime > n!+1 such that p - n! is also prime; then a(n) = p - n!
>
> The other day Bob Wilson sent me a file called a037153.txt,
> along with the following link:
>
> %H A037153 Robert G. Wilson v (rgwv at rgwv.com), <a href="a037153.txt">Table of n, a(n) for n=1..120000</a> a(n) is the least prime such that n! + p is also prime. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 02 2010]
>
> The file looks like this:
>
> # This is the a037153.txt text file.
> # A037153: a(n) is the least prime such that n! + p is also prime.
> # Revised dated 02 March 2010.
>   1       2
>   2       3
>   3       5
>   4       5
> ...
> 1197    3643
> 1198    1619
> 1199    8599
> 1200    5393
>
> There are several problems with this.  1.  The file has only 1200 lines,
> not 120000.  2. the statement
> # A037153: a(n) is the least prime such that n! + p is also prime.
> is wrong.
>
> My guess is that this is a b-file for the sequence S.  But there is a Mma
> program in A037153 which does indeed produce the true A037153.
> So I'm confused.
>
> So I wonder if some sequence fan could produce a b-file for A037153.
> Then either we will be able to update this remark:
>
> %C A037153 Analogous to Fortunate numbers and like them, the entries appear to be primes. In fact, the first 541 terms are primes. Are all terms prime?
>
> or we will get a counterexample, and we will have a new sequence S,
> and a third sequence giving indices where they are different!
>
> Bob, could you clarify what you did?
>
> Neil
>
>
>
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`