a(27) in A059097: A new lower higher bound

Jack Brennen jb at brennen.net
Mon Nov 21 23:16:21 CET 2005

David Wilson wrote:
> I wrote a Perl program to check A059097, and in the time it has taken me
> to write this message, it has confirmed that A059097 is complete up to
> 7,700,000.  Note that Perl is not particularly efficient at number crunching.

A program in C has taken about 4-5 minutes to find that there are no
other C(2n,n) which are (odd square)-free less than 2^31.

And yes, this sequence must be finite, beyond any reasonable doubt.
The density of n such that C(2n,n) is not divisible by p^2 is zero,
for any given odd prime p.  So this sequence in question, of which
26 elements are known, is the union of an infinity of sequences each
of which has density 0.  Since any two of the "p-sequences" seem
to be "orthogonal", I would assume that the 26 elements listed in the
OEIS represent the complete sequence.


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