Python 欧拉函数（Euler’s Totient Function），也称为简单欧拉函数（Euler’s Totient），是一个非常常用的数论函数。其定义为：对于正整数 $n$，欧拉函数 $\varphi(n)$ 表示小于等于 $n$ 的正整数中与 $n$ 互质的数的个数。互质的定义为两个数的最大公约数为 $1$。
例如，$\varphi(8) = 4$，因为小于 $8$ 的与 $8$ 互质的数有 $1, 3, 5, 7$。
欧拉函数有着广泛的应用，尤其是在密码学中，RSA公钥加密算法就是以欧拉函数为基础。
以下是使用 Python 中 sympy 库计算欧拉函数的完整攻略：
- 安装
sympy库。
!pip install sympy
- 引入
sympy库中的欧拉函数totient。
from sympy import totient
- 调用
totient(n)函数计算欧拉函数，参数n为正整数，返回值为与n互质的正整数的个数。
print(totient(8)) # 输出 4
以下是两个代码示例：
- 计算 $1$ 到 $100$ 中所有正整数的欧拉函数。
from sympy import totient
for i in range(1, 101):
print(f"phi({i}) = {totient(i)}")
输出：
phi(1) = 1
phi(2) = 1
phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4
phi(9) = 6
phi(10) = 4
phi(11) = 10
phi(12) = 4
phi(13) = 12
phi(14) = 6
phi(15) = 8
phi(16) = 8
phi(17) = 16
phi(18) = 6
phi(19) = 18
phi(20) = 8
phi(21) = 12
phi(22) = 10
phi(23) = 22
phi(24) = 8
phi(25) = 20
phi(26) = 12
phi(27) = 18
phi(28) = 12
phi(29) = 28
phi(30) = 8
phi(31) = 30
phi(32) = 16
phi(33) = 20
phi(34) = 16
phi(35) = 24
phi(36) = 12
phi(37) = 36
phi(38) = 18
phi(39) = 24
phi(40) = 16
phi(41) = 40
phi(42) = 12
phi(43) = 42
phi(44) = 20
phi(45) = 24
phi(46) = 22
phi(47) = 46
phi(48) = 16
phi(49) = 42
phi(50) = 20
phi(51) = 32
phi(52) = 24
phi(53) = 52
phi(54) = 18
phi(55) = 40
phi(56) = 24
phi(57) = 36
phi(58) = 28
phi(59) = 58
phi(60) = 16
phi(61) = 60
phi(62) = 30
phi(63) = 36
phi(64) = 32
phi(65) = 48
phi(66) = 20
phi(67) = 66
phi(68) = 32
phi(69) = 44
phi(70) = 24
phi(71) = 70
phi(72) = 24
phi(73) = 72
phi(74) = 36
phi(75) = 40
phi(76) = 36
phi(77) = 60
phi(78) = 24
phi(79) = 78
phi(80) = 32
phi(81) = 54
phi(82) = 40
phi(83) = 82
phi(84) = 24
phi(85) = 64
phi(86) = 42
phi(87) = 56
phi(88) = 40
phi(89) = 88
phi(90) = 24
phi(91) = 72
phi(92) = 44
phi(93) = 60
phi(94) = 46
phi(95) = 72
phi(96) = 32
phi(97) = 96
phi(98) = 42
phi(99) = 60
phi(100) = 40
- 将多个正整数的欧拉函数求和。
from sympy import totient
n_list = [3, 5, 7, 10]
total = sum([totient(n) for n in n_list])
print(total) # 输出 11
其中，sum() 函数用于将多个欧拉函数的返回值相加。