[seqfan] oeis.org/search?q=2014,3,14

[M. F. Hasler]

Dear SeqFans,

the first suggestion on https://oeis.org/wiki/Suggested_Projects is to
add sequences for the month of March, which happens to be now (while
that "call to arms" dates back to 2011) .
Today I did so for March 30, with a sequence starting 3,30,.... This
led to a search for composite numbers which have all their divisors as
substring in their decimal expansion. Charles Greathouse found the
second such number (after 125) to be A239058(870356)=17692313, the
next larger one is still unknown!

Then I noticed that there was still no SOTD for tomorrow, March 10.
The only search hit for 3, 10, __,2014 is
A132995 = gcd of sum and product of the first n primes.
It happens that the 2014 is immediately followed by 3,14:
http://oeis.org/search?q=2014,3,14
which would have made a nice candidate for this year's pi day.
But that day already has a proposal for a pi related sequence,
so I kept this as SOTD for March 10 and added a b-file.
I just wanted to invite you to admire the nice resulting graph,
https://oeis.org/A132995/graph

Is it reasonable to conjecture that there are infinitely many a(n)=1,
a(n)=2, a(n)=3?
Is there some, or maybe even infinitely many, a(n)=P for any given
product P of distinct primes, i.e., any squarefree number P?
Can someone prove these conjectures?

Have a nice week,

Maximilian