[seqfan] ArcCoth Irreducible Numbers

[Paul D Hanna]

    I will try to answer the question raised in my prior e-mail.

    The sequence is the "inverse hyperbolic cotangent 
irreducible numbers", and begins with: 
{1,2,3,4,6,8,10,12,14,16,18,20,22,24,28,30,32,36,...}

    Below is a definition, some properties and examples.

    Is this the same as A045718 (Nearest-neighbors of primes)?

    Regards,
         Paul

DEFINITION.
Sequence of positive integers such that the arctanh 
of these numbers form a basis for the space of 
arctanh of rationals > 1.

Let h(x,y)=(xy-1)/(y-x).  

If n is in the sequence, then n can not be formed 
by  h(x,y) for any y < 2*n  
where y>x and x and y are both positive integers.

PROPERTIES.
* only odds are: 1,3
   since (2n+1) = h( n, (2n-1))  for n>1.

* all evens except: 
   {26,34,56,64,76,86,94,...} (any pattern?)

EXAMPLES.
5=h(2,3)
7=h(3,5)
9=h(4,7)
11=h(3,4)=h(5,9)
13=h(5,8)=h(6,11)
15=h(7,13)
17=h(8,15)
19=h(4,5)=h(9,17)=h(7,11)
21=h(10,19)
23=h(7,10)=h(11,21)
25=h(9,14)=h(12,23)
26=h(11,19)
27=h(13,25)
29=h(5,6)=h(8,11)=h(9,13)=h(14,27)
31=h(7,9)=h(11,17)=h(15,29)
33=h(16,31)
34=h(13,21)
...
56=h(23,29)
64=h(19,27)=h(25,41)=h(29,53)
76=h(21,29)
86=h(35,59)
94=h(37,61)
...
On Fri, 6 Jun 2003 18:20:36 GMT pauldhanna at juno.com writes:
> ... the hyperbolic version of Stormer numbers 
> ("hyperbolic Stormer numbers", say), such that the 
> arctanh of these numbers form a basis for the space of 
> arctanh of rationals > 1? 
> If we permit 1 to be a hyperbolic Stormer number (though a 
> singularity), the sequence would begin 1,2,3,...
> 
> If non-trivial, what is the rest of the sequence of 'hyperbolic 
> Stormer numbers'?  
>      Paul
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