A036471

[kohmoto]

    Hi, Seqfans

    An exhaustive search of A036471  with a computer up to 10^7 is almost 
impossible.

    So, using my algorithm, I searched small examples which are “regular 
type”.



    See this site, there is an explanation of "type".



              http://amicable.homepage.dk/apstat.htm#typesys







    I found these examples.



    2^9*3^2*13*31*(5*11,71)*(7*23,191) 
11 digits

    2^15*3*5^2*11*31*257*(1439,19*71,23*59,29*47) 
16 digits

    2^3*3^4*5^2*(7*23,191)*(17*19,359) 
9 digits

    (2^3*19*41,2^5*199)*(3*5*7*13,3^2*5*7*139) 
8 digits

    (2^3*19*41,2^5*199)*(3^3*5*7*71,3^3*5*17*31) 
9 digits

    (2^3*19*41,2^5*199)*(3^2*7*13*5*17,3^2*7*13*107) 
9 digits

    2^7*3^2*13*(5*11,71)*(7*59,479) 
9 digits

    2^5*3^4*5^2*(17*19,359)*(23*29,719) 
11 digits







    Where, x*(y,z) means (x*z,x*y). (x,y)*(z,u)=(x*z,x*u,y*z,y*z)





    I think that if a smaller one than 3270960  exists, then it must be 
“irregular".



    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}.

    The numbers x,y,z,u of small Amicable Quadruple must satisfy the 
following conditions.

             .

    o Sigma(x)=Sigma(y) and Sigma(z)=Sigma(u), or 
Sigma(x)=Sigma(y)=Sigma(z)=Sigma(u)

    o all prime factors of x,y,z,u are small

    o all prime factors of x+y and z+u are small, or all  prime factors of 
x+y+z+u are small



    For example, if {x,y}={7*23,191} and {z,u}={17*19,359} then

    o Sigma(7*23)=Sigma(191), Sigma(17*19)=Sigma(359)

    o all prime factors of 7*23,191,17*19,359  are small

    o x+y=2^5*11, z+u=2*11*31

    So, (7*23,191)*(17*19,359) generates 2^3*3^4*5^2*(7*23,191)*(17*19,359)





    If we obtain all small seeds then we will calculate all small regular 
terms of A036471.



    Yasutoshi


-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://list.seqfan.eu/pipermail/seqfan/attachments/20061114/6c8412c2/attachment-0001.htm>