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Problem 342


Problem 342


The totient of a square is a cube

Consider the number 50.
502 = 2500 = 22 × 54, so φ(2500) = 2 × 4 × 53 = 8 × 53 = 23 × 53. 1
So 2500 is a square and φ(2500) is a cube.

Find the sum of all numbers n, 1 < n < 1010 such that φ(n2) is a cube.

1 φ denotes Euler’s totient function.


平方数的总计函数为立方数

考虑数50。
502 = 2500 = 22 × 54,因此φ(2500) = 2 × 4 × 53 = 8 × 53 = 23 × 531
所以2500是一个平方数,而φ(2500)是一个立方数。

有些数1 < n < 1010满足φ(n2)为立方数,求所有这类数的数目。

1 φ表示欧拉总计函数