A bracelet is made by connecting at least three numbered beads in a circle. Each bead can only display , , or any number of the form or for odd prime .
In addition a magic bracelet must satisfy the following two conditions:
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 be the sum of the potency of each magic bracelet which can be formed using positive integers not exceeding , where rotations and reflections of an arrangement are considered equivalent. You are given and .
Find .
一个数镯由至少三个标有数字的珠子和连接它们的环构成。数镯上的珠子的标号只能是 、、、 四种形式之一(其中 为任意奇质数, 为任意正整数)。
进一步,定义魔法数镯为满足以下条件的数镯:
对任意魔法数镯,定义其魔法效力为其所有珠子标号之和。上图中的魔法数镯共有 5 个珠子,魔法效力为 155。
记 为:满足所有珠子标号均为不超过 的正整数的魔法数镯的魔法效力之和,其中,经旋转或翻转后相同的数镯视为相同的数镯。已知:、。
求 。
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