<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2716.2200" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV></DIV>
<DIV> I will try to answer the question raised in my
prior e-mail.</DIV>
<DIV> </DIV>
<DIV> The sequence is the "inverse hyperbolic cotangent </DIV>
<DIV>irreducible numbers", and begins with:
<BR>{1,2,3,4,6,8,10,12,14,16,18,20,22,24,28,30,32,36,...}<BR></DIV>
<DIV> Below is a definition, some properties and
examples.</DIV>
<DIV> </DIV>
<DIV> Is this the same as <STRONG>A045718</STRONG>
(Nearest-neighbors of primes)?</DIV>
<DIV> </DIV>
<DIV> Regards,</DIV>
<DIV> Paul</DIV>
<DIV> </DIV>
<DIV>DEFINITION.</DIV>
<DIV>Sequence of positive integers such that the arctanh </DIV>
<DIV>of these numbers form a basis for the space of </DIV>
<DIV>arctanh of rationals > 1.</DIV>
<DIV><BR>Let h(x,y)=(xy-1)/(y-x). </DIV>
<DIV> </DIV>
<DIV>If n is in the sequence, then n can not be formed </DIV>
<DIV>by h(x,y) for any y < 2*n </DIV>
<DIV>where y>x and x and y are both positive integers.</DIV>
<DIV> </DIV>
<DIV>PROPERTIES.<BR>* only odds are: 1,3<BR> since (2n+1) = h( n,
(2n-1)) for n>1.</DIV>
<DIV> </DIV>
<DIV>* all evens except: <BR> {26,34,56,64,76,86,94,...} (any
pattern?)</DIV>
<DIV> </DIV>
<DIV>EXAMPLES.<BR>5=h(2,3)<BR>7=h(3,5)<BR>9=h(4,7)<BR>11=h(3,4)=h(5,9)<BR>13=h(5,8)=h(6,11)<BR>15=h(7,13)<BR>17=h(8,15)<BR>19=h(4,5)=h(9,17)=h(7,11)<BR>21=h(10,19)<BR>23=h(7,10)=h(11,21)<BR>25=h(9,14)=h(12,23)<BR>26=h(11,19)<BR>27=h(13,25)<BR>29=h(5,6)=h(8,11)=h(9,13)=h(14,27)<BR>31=h(7,9)=h(11,17)=h(15,29)<BR>33=h(16,31)<BR>34=h(13,21)<BR>...<BR>56=h(23,29)<BR>64=h(19,27)=h(25,41)=h(29,53)<BR>76=h(21,29)<BR>86=h(35,59)<BR>94=h(37,61)<BR>...</DIV>
<DIV>On Fri, 6 Jun 2003 18:20:36 GMT <A
href="mailto:pauldhanna@juno.com">pauldhanna@juno.com</A> writes:<BR>> ...
the hyperbolic version of Stormer numbers </DIV>
<DIV>> ("hyperbolic Stormer numbers", say), such that the </DIV>
<DIV>> arctanh of these numbers form a basis for the space of </DIV>
<DIV>> arctanh of rationals > 1? <BR>> If we permit 1 to
be a hyperbolic Stormer number (though a <BR>> singularity), the sequence
would begin 1,2,3,...<BR>> <BR>> If non-trivial, what is the rest of the
sequence of 'hyperbolic <BR>> Stormer numbers'?
<BR>> Paul</DIV></BODY></HTML>