Question on Sequence A046907 from Sloane's Database

[David W. Wilson]

"James A. Sellers" wrote:
> 
> Dear Members of SeqFan,
> 
> I have only been a member of this list for a few days, so forgive me if this is an inappropriate message to send!!
> 
> I have a question on sequence A046907, which gives the values of n for which binomial(2n-1, n) is squarefree.   The sequence is 1,2,3,4,6,9,10,12,36.  I have now tried n=37 up to 1500 and have not found another term for the sequence.  Is this sequence known to be finite??  That would be an interesting result, if it is known.
> 
> Thanks for your time!  I hope to hear from some of you soon.
> 
> James
> 
> ****************************************
> James A. Sellers
> Associate Professor, Mathematics
> Cedarville College
> 
> sellersj at cedarville.edu
> http://www.cedarville.edu/dept/sm/jas_www.htm

This is one a broad set of problems which devolve to a unsolved general
problem.  The general problem is:

Is the intersection of a given a-automatic and b-automatic set finite?

A k-automatic set is a set of integers whose base-k representations form
a regular language, that is, a language accepted by a finite automaton
or state machine.

If bases a and b are incompatible (do not have a common power), and if
an a-automatic set Sa and b-automatic set Sb are both of density 0
over the integers, it is believed that Sa intersect Sb is finite
(or something along this line).

The current problem is whether f(n) = (2n-1 choose n) is squarefree for
a finite number of n.  It turns out that f(n) is divisible by 4 unless
n belongs to a 2-automatic set S2 (which happens to be the set of
numbers
whose binary representations contain at most two 1's).  Similarly, f(n)
is divisibly by 9 unless n belongs to a 3-automatic set S3 (much harder
to describe).  If f(n) is squarefree, therefore, n must belong to
S = S2 intersect S3.  It is very probable that S is finite, but no
proof is known.

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There are other prize problems that devolve to the same general problem.
For instance:

Are there infinitely many positive integers n such that 2^n does not
contain a 7 in its decimal expansion?

The powers of 2 form a 2-automatic set S2.  The numbers which do not
contain a 7 form a 10-automatic set S10.  The problem devolves to
whether S2 intersect S10 is finite.

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Another prize problem:

Are there infinitely many positive integers n such that 2n choose n
is divisible by neither 3, 5, nor 7?

Here, the n with (2n choose n) indivisible by 3, 5, and 7, respectively,
form sets S3, S5, and S7 which are respectively 3-automatic,
5-automatic,
and 7-automatic.  We are asking if S3 intersect S5 intersect S7 is
finite.  In all likelyhood, an interesection of any two of these is
finite.