koh zbi74583.boat at orange.zero.jp
Mon Aug 25 02:55:07 CEST 2008

```    Hi,Seqfans

I considered about the regular type of solution of Quadratic Amicable Number.
UnitarySigma(x) = UnitarySigma(y) = 1/k*(x^2+m*y^2)/(x+y)
k<x<y

If
x=c*u
y=c*v
GCD(c,u)=1, GCD(c,v)=1
Then c is called "Sprout", {u,v} is called "Seed".
{x,y} is called "Regular type".

k=((u-v)*(1-v)+i*v^2)*c/UnitarySigma(c)
m=(u-v)*(u+v)-1 + i*(u+v)*(1+v)

c ｍust satisfy the following conditions.

(u-v)*(1-v)+i*v^2 = 0 Mod UnitalySigma(c) , 0<=i
(u-v)*(1-v)+i*v^2 < c*u
GCD(c,u)=1, GCD(c,v)=1

So, {number of the sprout of seed {u,v} for eadh i} < A141059(((u-v)*(1-v)+i*v^2)

{u,v}={6,11} It is the smallest seed.

c = 7^2,419....

{u,v)={10,17}

c = 13,3^3,137,229,....

{u,v}={12,19}

c = 41,5^3,....

{u,v}={15,23}

c = 43,....

{u,v}={18,29}

c = 43,307,....

{u,v]={20,29]

c = 17,3^3,41,251,....

{u,v}={21,31}

c = 19,2^2*5,29,2*19,2^2*11,59,2^2*29,149,2^2*59,....

x :
130,270,294,340,399,420,492,540,609,645, 774, 798, 820, 924,1239,1370

y :
221,459,539,493,589,620,779,783,899,989,1247,1178,1189,1364,1829,2328
It is difficult to get irregular type by hand.
Could anyone search them using a computer?

To Neil

Do x,y fit to OEIS?

Yasutoshi

```