[seqfan] Re: A_k and RMPN

[Charles Greathouse]

I notice that the 2-tuples are already ion the OEIS as A206708. Perhaps an
index entry would also be warranted?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Sun, Dec 8, 2013 at 8:04 PM, <zbi74583.boat at orange.zero.jp> wrote:

>     Hi,Seqfans
>
>     I am the first mathematician who computed Amicable k-ple for k=4, k=5.
>
>     http://mathworld.wolfram.com/AmicableQuadruple.html
>
>     http://list.seqfan.eu/pipermail/seqfan/2008-November/000217.html
>
>     Still for 5 < k, no Amicable k-ple is known.
>
>     OEIS has Amicable 4-ple. A036371-A036474
>
>     But it does not yet have A_5.
>     I think OEIS should have it.
>
>     Could anyone compute the first several terms?
>
>     If you have a good algorithm for computing RMPN then it is easy to
> compute
> rapidly A_k.
>     I explain the method to compute Amicable k-ple.
>
>     [ How to compute A_k ]
>
>     Compute Sigma(m) 1<=m<=n , List them
>
>     Sort the list by the order of "<" , List them as S_i 1<=i<=n
>
>     Find k-ple of S_i j<=i<=j+k-1 , S_i=S
>
>     Let Sigma(m_i)=S 1<=i<=k
>     Let x_i=c*m_i    1<=i<=k  , GCD(c,m_i)=1 .... E1
>
>     If x_i is Amicable k-ple Then
>     Sigma(x_i)=Sum_{1<=r<=k} x_r    1<=i<=k .... E0
>
>     From E0,E1
>     Sigma(c*m_i)=Sum_{1<=r<=k} c*m_r    1<=i<=k .... E2
>     Hence
>     Sigma(c)*Sigma(m_i)=c*(Sum_{1<=r<=k} m_r)
>     Sigma(c)=u*c , u=(Sum_{1<=r<=k} m_r)/Sigma(m_i)
>     c is RMPN
>     Compute c
>
>     If GCD(c,m_i)=1 1<=i<=k then (x_i) 1<=i<=k is Amicable k-ple
>     If m_i has no small prime factor  for all i and all prime factors of
> Sum_i
> m_i are small then the probability of success of computing c is high.
>
>     I named {m_i} "Seed" and named c "Spout".
>
>     Example of smooth Seed of A_2.
>
>     http://amicable.adsl.dk/aliquot/apstat/apco30.txt
>
>     Seed of 21 Kohmoto 1997 193D =
>     {89*100329964009286143948575662850542265921787709 ,
> 9029696760835752955371809656548803932960893899}
>
>     Sum_i m_i = 2^64*3^29*5^11*7^4*11^2
>
>
>
>     Yasutoshi
>
>
>
>
>
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