# [seqfan] A_k and RMPN

zbi74583.boat at orange.zero.jp zbi74583.boat at orange.zero.jp
Mon Dec 9 02:04:11 CET 2013

```    Hi,Seqfans

I am the first mathematician who computed Amicable k-ple for k=4, k=5.

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.

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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