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<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3> Hi, Seqfans</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3> An exhaustive search of A036471
with a computer up to 10^7 is almost impossible.</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>So, using my algorithm,
I searched small examples which are “regular type”.</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"><FONT face=Century
size=3></FONT></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"><FONT face=Century size=3> See this
site, there is an explanation of "type".</FONT></SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"><FONT face=Century
size=3></FONT></SPAN></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><SPAN
style="mso-spacerun: yes"><FONT face=Century
size=3>
<A
href="http://amicable.homepage.dk/apstat.htm#typesys">http://amicable.homepage.dk/apstat.htm#typesys</A></FONT></SPAN></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><?xml:namespace
prefix = o ns = "urn:schemas-microsoft-com:office:office" /><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN
lang=EN-US><o:p></o:p></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN
lang=EN-US><o:p> </o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> I found these
examples.</SPAN></FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"></SPAN></FONT></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN></FONT></FONT></SPAN><SPAN lang=EN-US><FONT size=3><FONT
face=Century>2^9*3^2*13*31*(5*11,71)*(7*23,191)<SPAN
style="mso-spacerun: yes">
</SPAN>11 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>2^15*3*5^2*11*31*257*(1439,19*71,23*59,29*47)<SPAN
style="mso-spacerun: yes">
</SPAN>16 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>2^3*3^4*5^2*(7*23,191)*(17*19,359)<SPAN
style="mso-spacerun: yes">
</SPAN>9 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>(2^3*19*41,2^5*199)*(3*5*7*13,3^2*5*7*139)<SPAN
style="mso-spacerun: yes">
</SPAN>8 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>(2^3*19*41,2^5*199)*(<SPAN
lang=EN-US>3^3*5*7*71,3^3*5*17*31)</SPAN><SPAN
style="mso-spacerun: yes">
</SPAN>9 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>(2^3*19*41,2^5*199)*(3^2*7*13*5*17,3^2*7*13*107)<SPAN
style="mso-spacerun: yes">
</SPAN>9 digits</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3>
2^7*3^2*13*(5*11,71)*(7*59,479) 9
digits</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3>
2^5*3^4*5^2*(17*19,359)*(23*29,719)
11 digits</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
face=Century size=3> </FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> <FONT face="MS UI Gothic" size=2>
</FONT></FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes"> </SPAN>Where, x*(y,z) means
(x*z,x*y). (x,y)*(z,u)=(x*z,x*u,y*z,y*z)<SPAN
style="mso-spacerun: yes"> </SPAN></FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN>I think that if a smaller one than<SPAN
style="mso-spacerun: yes"> 3270960 </SPAN>exists, then it must be
“irregular".<SPAN
style="mso-spacerun: yes">
</SPAN></FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN>Regurer type Amicable Quadruple are generated from “seeds” which are two
pairs or one quadruple of primes or almost primes, {x,y} and {z,u} or
{x,y,z,u}.</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN>The numbers x,y,z,u of small Amicable Quadruple must satisfy the
following<SPAN
style="mso-spacerun: yes"> conditions.</SPAN></FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN
style="mso-spacerun: yes">
</SPAN>.</FONT></FONT></SPAN></P><SPAN lang=EN-US><FONT size=3><FONT
face=Century><SPAN style="mso-spacerun: yes">
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN>o Sigma(x)=Sigma(y) and Sigma(z)=Sigma(u), or
Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>o all prime factors of x,y,z,u are
small</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>o all prime factors of x+y and z+u are
small, or all </FONT></FONT></SPAN><SPAN lang=EN-US><FONT size=3><FONT
face=Century><SPAN style="mso-spacerun: yes"> </SPAN>prime factors of
x+y+z+u are small</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>For example, if {x,y}={7*23,191} and
{z,u}={17*19,359} then </FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>o Sigma(7*23)=Sigma(191),
Sigma(17*19)=Sigma(359)</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN style="mso-spacerun: yes"> o
all prime factors of 7*23,191,17*19,359 are small</P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN></FONT></FONT></SPAN></SPAN></FONT></FONT></SPAN><SPAN
lang=EN-US><FONT face=Century size=3>o x+y=2^5*11, z+u=2*11*31</FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes">
</SPAN>So, (7*23,191)*(17*19,359) generates<SPAN
style="mso-spacerun: yes"> 2^3*3^4*5^2*(7*23,191)*(17*19,359)<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN></FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face="MS UI Gothic" size=2><SPAN style="mso-spacerun: yes"><SPAN
style="mso-spacerun: yes"></SPAN></SPAN></FONT></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><o:p><FONT
face=Century size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>If we obtain<SPAN
style="mso-spacerun: yes"> </SPAN>all small seeds then we<SPAN
style="mso-spacerun: yes"> </SPAN>will calculate all small regular terms of
A036471.</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face="MS UI Gothic" size=2></FONT></FONT></SPAN> </P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN lang=EN-US><FONT
size=3><FONT face=Century> Yasutoshi</FONT></FONT></SPAN></P>
<P class=MsoNormal style="MARGIN: 0mm 0mm 0pt"><SPAN
lang=EN-US>
</SPAN></P></SPAN></FONT></FONT></SPAN></FONT></DIV></BODY></HTML>