[seqfan] Sequence A037153

[N. J. A. Sloane]

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