Magic Bracelets (noch nicht übersetzt)

A bracelet is made by connecting at least three numbered beads in a circle. Each bead can only display $1$, $2$, or any number of the form $p^k$ or $2p^k$ for odd prime $p$.

In addition a magic bracelet must satisfy the following two conditions:

• no two beads display the same number
• the product of the numbers of any two adjacent beads is of the form $x^2+1$

Define the potency of a magic bracelet to be the sum of numbers on its beads.

The example is a magic bracelet with five beads which has a potency of 155.

Let $F(N)$ be the sum of the potency of each magic bracelet which can be formed using positive integers not exceeding $N$, where rotations and reflections of an arrangement are considered equivalent. You are given $F(20)=258$ and $F(10^2)=538768$.

Find $F(10^6)$.