# [seqfan] A07421 real world geometry

Daniel Palasek dan.palasek at gmail.com
Wed May 13 20:00:47 CEST 2020

```A07421 is defined as "Squares satisfying sigma(n)==0 (mod 3)", which is not
clear how that connects to this:
It appears to have an interesting property with real world geometry.
The sequence is perfect squares that are themselves the sums of 3 perfect
squares of unequal size:

*49, 169, 196, 361 ...*
The 3 unequal squares are the integer radius solutions of Sangaku 3
touching circles sharing a tangent line https://en.wikipedia.org/wiki/Sangak

*49=7^2*

*49=36+9+4 (a+b+c)*
*     or 6^2 + 3^2 + 2^2*
* 1/sqrt(c) = 1/sqrt(a) + 1/sqrt(b) *

*1/sqrt(4) = 1/sqrt(36) + 1/sqrt(9)*

*i.e. circles of radius 36, 9, 4 will touch when aligned on a tangent.*

*169=13^2*
*169=144+16+9*
*     or 12^2 + 4^2 + 3^2*

*1/sqrt(9) = 1/sqrt(144) + 1/sqrt(16) *

*196=14^2*
*196=144+36+16*
*     or 12^2 + 6^2 + 4^2*

*1/sqrt(16) = 1/sqrt(144) + 1/sqrt(36) *

*361=19^2*
*361=225+100+36 *
*     or 15^2 + 10^2 + 6^2*
*1/sqrt(36) = 1/sqrt(225) + 1/sqrt(100) *

-Daniel
```