[seqfan] Re: min(d(p^n-1)) for sufficiently large p

Max Alekseyev maxale at gmail.com
Thu May 27 20:07:06 CEST 2010

I think the current definition is imprecise in two ways: first, it
sounds like the bound a(n) for d(p^n - 1) may depend on how large the
prime p is; second, it is not said that the bound is tight.
I would suggest to define the sequence via the limit inferior "lim
inf" notation to make it precise. E.g.:

a(n) = lim inf A000005(p^n - 1) as p tends to infinity over the primes


On Thu, May 27, 2010 at 7:51 AM,  <hv at crypt.org> wrote:
> %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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