The Leech lattice is related to the formula:
$$1^2+2^2+3^2+....+24^2=70^2$$
This is related in turn to 26D bosonic string theory.
Is there another known formula that involves the integers 1..8, which is related perhaps to $E_8$ or 10D supergravity? At first I thought it might be the sums of squares of the first 8 odd numbers but this didn't give anything interesting. (Because supersymmetry includes half spin particles $1/2,3/2,5/2$ etc.
(The Leech lattice and $E_8$ being the lattices with maximal sphere packing in their respective dimensions).
Edit: I found another examples. Not sure if they have any relevance: $$2^2+5^2+8^2+11^2+14^2+17^2+20^2+23^2+26^2=48^2$$ Which goes up in 3's and also this is 9 terms. Probably just a coincidence.