MAX(p_A093407)

[hv at crypt.org]

Robert, 
thanks a lot!

I included your explanation as %C -
is it OK?
zak

Revised v:

%I A000001
%S A000001
2,3,4,7,16,31,127,256,8191,65536,131071,524287,2147483647,2305843009213693951,618970019642690137449562111,162259276829213363391578010288127,170141183460469231731687303715884105727
%N A000001 Numbers n such that
lcm(1..(n-1))<lcm(1..n)<lcm(1..(n+1)).
%C A000001 Or, numbers n such that
A003418(n-1)<A003418(n)<A003418(n+1).
Sequence is the union(A019434 - 1, A000668).
%C  lcm(1..n-1) < lcm(1..n) iff n is a prime power. So
the sequence consists of those n for which both n and
n+1 are prime powers.  By Catalan's conjecture (proved
by Mihailescu), the only case where n and n+1 are both
powers > 1 is n=8.  Otherwise, whichever of n and n+1
is even must be a power of 2, and the other must be a
prime: either a Mersenne prime if n+1 is the power of
2, or a Fermat prime if n is the power of 2. - Robert
Israel (israel at math.ubc.ca) 
%Y A000001 A000668, A003418, A019434, 
%O A000001 1
%K A000001 ,nice, nonn
%A A000001 Zak Seidov (zakseidov at yahoo.com), Jan
 17 2008


--- Robert Israel <israel at math.ubc.ca> wrote:

> lcm(1..n-1) < lcm(1..n) if and only if n is a prime
> power.
> So your sequence consists of those n for which both
> n and n+1
> are prime powers.  By Catalan's conjecture (proved
> by
> Mihailescu), the only case where n and n+1 are both
> powers > 1
> is n=8.  Otherwise, whichever of n and n+1 is even
> must be a
> power of 2, and the other must be a prime: either a
> Mersenne
> prime if n+1 is the power of 2, or a Fermat prime if
> n is the
> power of 2.
> 
> Robert Israel                               
> israel at math.ubc.ca
> Department of Mathematics       
> http://www.math.ubc.ca/~israel 
> University of British Columbia            Vancouver,
> BC, Canada
> 
> On Wed, 16 Jan 2008, zak seidov wrote:
> 
> > %I A000001
> > %S A000001 2,3,4,7,8,16,31,127,256
> > %N A000001 Numbers n such that
> > lcm(1..(n-1))<lcm(1..n)<lcm(1..(n+1)).
> > %C A000001 Or, numbers n such that
> > A003418(n-1)<A003418(n)<A003418(n+1).
> > Are all numbers of kind 2^m or 2^m-1?
> > %t A000001 ta=Table[(LCM @@ Range[n]),{n,2000}];
> >
>
Do[If[ta[[i-1]]<ta[[i]]<ta[[i+1]],Print[i]],{i,2,1999}]
> > %Y A000001 A003418
> > %O A000001 1
> > %K A000001 ,nonn,
> > %A A000001 Zak Seidov (zakseidov at yahoo.com), Jan
> 17
> > 2008
> >
> >
> >
> >     
>
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