two integrals and a question about prime trinomials

[David Wilson]

Empirically it appears that A002426(n) is divisible by 3 exactly when
the base-3 representation of n includes the digit 2.  A similar
phenomenon happens with divisor 7.  A002426(n) appears to be
divisible by 7 exactly when the base-7 representation of n includes
the digit 3.  I would not be surprised if similar things happen
for other prime divisors.

The n for which A002426(n) is divisible by 3 forms a 3-automatic set,
and the analogous statement is true for 7.   If this generalizes to the
other primes, then what we seek is a proof that the union of p-automatic
sets with incompatible bases p is cofinite, such questions are to my
knowledge unanswerable at the present time.

BTW, given that the above observations about divisors 3 and 7 are
indeed true, Wilson's conjecture of recent messages would imply that
all but a finite number of elements of A002426(n) are divisible by 3
or 7.  This doesn't give much purchase in showing that A002426(n)
is composite for n > 4, however, since we have no idea how big the
largest offending n is, and empirically it looks rather large.

----- Original Message ----- 
From: "Eric W. Weisstein" <eww at wolfram.com>
To: <ham>; "Sequence Fans Mailing List" <seqfan at ext.jussieu.fr>
Sent: Thursday, February 24, 2005 2:43 PM
Subject: two integrals and a question about prime trinomials


>I anyone able to get the following two integrals in closed form?
>
> Integrate[Exp[-2x^2]Erf[x]^2,{x,0,Infinity}]
> Integrate[Exp[-x^2]Erf[x]^3/x,{x,0,Infinity}]
>
> (I'm not.)
>
> Also, Jonathan Vos Post and Robert G. Wilson have noted that there are a
> dearth of prime central trinomial coefficients
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A002426
> after n=4.  In fact, I just did a quick check and there are no others
> besides n=2, 3, 4 up to n = 12,000.  Can someone prove that A002426(n) is
> actually composite for all n>4?
>
> Cheers,
> -Eric
>
>
>
>
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