[seqfan] Re: min(d(p^n-1)) for sufficiently large p
franktaw at netscape.net
franktaw at netscape.net
Thu May 27 22:47:05 CEST 2010
I think I would recommend something like "a(n) is the largest value m
such that p^n-1 has at least m divisors for sufficiently large primes
p", with a comment like "All terms in this sequence are conjectured;
even that such a value exists is a conjecture." You ought to be able to
find a suitable conjecture to support the values on-line somewhere -
perhaps at <http://www.primepuzzles.net/conjectures/>.
I don't normally support making a sequence with all conjectured values,
but in this case, this really is what the sequence is. If someone
found, for example, that there are only finitely many Sophie Germain
primes, the values in the sequence would be changed.
Do you have a program that generates these values? If you do, it should
definitely be included - send it as a separate file with a link if it
is too long for direct inclusion. Otherwise some more complete
description of how the values were determined ought to be included.
Franklin T. Adams-Watters
-----Original Message-----
From: hv at crypt.org
%I A000001
%S A000001
4,32,8,160,8,384,8,384,16,256,8,7680,8,128,32,1792,8,4096,8,3840,32,
%T A000001
256,8,36864,16,128,32,2560,8,24576,8,4096,32,128,32,327680,8,128,32,
%U A000001
36864,8,18432,8,2560,128,256,8,344064,16,1024,32,2560,8,20480,32,
%N A000001 p^n-1 has at least a(n) divisors for all sufficiently large
primes p
%F A000001 a(n) = d(A079612(n)) . 2^d(n) (where d(n)=A000005(n))
%e A000001 From A079612() we know that 24 must divide p^2-1 for all
primes p
except 2 and 3. With a finite number of small exceptions, the factors
p-1 and
p+1 must contribute either an additional distinct prime factor or
enough small
repeated factors to ensure that d(p^2-1) >= d(24qr) with q, r distinct
primes >
3, so a(2) = d(24qr) = 32.
%Y A000001 Cf. A000005, A079612.
%K A000001 new,nonn
%O A000001 1,1
%A A000001 hv at crypt.org (Hugo van der Sanden)
Is it reasonable to name this sequence in this way? I do not mean to
imply,
for example, that stating a(1)=4 means I have a proof of the infinitude
of
Sophie Germain primes, only that d(p-1) is known to be >=4 by well-known
algebraic and modular considerations.
If it is not reasonable, maybe the name should be changed to correspond
to
the formula and the old name moved to a comment. However that feels like
it would be a less satisfactory (and much less useful) name.
I welcome any other suggestions for improving this sequence.
Hugo
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