New sequence??

[Richard Guy]

No, that was oversimplified.  Let me try again:

It is the same as  A068527  for quite a while:

0, 0, 2, 1, 0, 4, 3, 2, 1, 0, 6, 5, 4, 3, 3, 1, 0,
8, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 8, 7, 6, 5, 4, 3,
2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,
14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,
15, 14, 14,     R.

On Fri, 2 Apr 2004, John Conway wrote:

> On Fri, 2 Apr 2004, Richard Guy wrote:
> 
> > Sorry about that!!  I wasn't including n.
> > 
> > Shift all my members one place.     R.
> 
> ALL your members??!!   JHC
> 
>  
> > On Thu, 1 Apr 2004, Richard Guy wrote:
> > 
> > > If  n = 26,  we want the least  t_n  so that some
> > > subset of  26 + 1, 26 + 2, ..., 26 + t_n  has a
> > > square product.  I make this to be
> > > 
> > >          27* 28 * 30 * 32 * 35     [...]
> 
>      I wonder how many sequencers are aware of the fact that
> for any sequence  a,b,c,...  of positive integers, the sequence
> a, a^b, a^b^c, a^b^c^d, ...  converges to any modulus?
> 
>     A remark of Jerrold Grossman's led me to discover this for
> myself, and it pleased me very much, because of course it makes
> infinite "numbers" like  10^10^10^...  meaningful modulo any number.
> For instance, that one is congruent to 38 modulo 47.  
> 
>    However, when I mentioned it to Dick Bumby last weekend, he said
> that this theorem had been proposed in the Monthly when he was Problems
> Editor, and had produced a lot of correspondence, some of which was
> from Jerry Grossman, and some of which gave early references, so the
> I'm far from being its first author.  Oh well!
> 
>    It gives rise to lots of sequences, of which my favorite is that
> giving the least positive remainders of  2^3^4^5^...  modulo 2,3,4,5,... .
> 
>     JHC
>