Comments

Susam Pal wrote:

Yes, it is quite straightforward to extend the solution to include numbers divisible by 7. The new indicator function can be defined like this: \[ I_7(n) = \frac{1}{7} \sum_{k = 0}^6 e^{2 \pi i k n/7} = \frac{1}{7} + \frac{2}{7} \cos \left( \frac{2 \pi n}{7} \right) + \frac{2}{7} \cos \left( \frac{4 \pi n}{7} \right) + \frac{2}{7} \cos \left( \frac{6 \pi n}{7} \right). \] The new Fourier series is going to be: \[ g(n) = I_3(n) + 2 \, I_5(n) + 4 \, I_7(n) = f(n) + 4 \, I_7(n). \] So we only need to add the new term \( 4 \, I_7(n) \) to \( f(n), \) which gives the new Fourier series \begin{align*} g(n) = \frac{137}{105} &+ \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right) + \frac{4}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{4}{5} \cos \left( \frac{4 \pi n}{5} \right) \\ &+ \frac{8}{7} \cos \left( \frac{2 \pi n}{7} \right) + \frac{8}{7} \cos \left( \frac{4 \pi n}{7} \right) + \frac{8}{7} \cos \left( \frac{6 \pi n}{7} \right). \end{align*} The indicators-based solution simply becomes:

from math import cos, pi
for n in range(1, 1001):
    s = [n, 'Fizz', 'Buzz', 'FizzBuzz', 'Jazz', 'FizzJazz', 'BuzzJazz', 'FizzBuzzJazz']
    i = (n % 3 == 0) + 2 * (n % 5 == 0) + 4 * (n % 7 == 0)
    print(s[i])

The cosine-based solution becomes:

from math import cos, pi
for n in range(1, 1001):
    s = [n, 'Fizz', 'Buzz', 'FizzBuzz', 'Jazz', 'FizzJazz', 'BuzzJazz', 'FizzBuzzJazz']
    i = round((137 / 105) + (2 / 3) * cos(2 * pi * n / 3)
                          + (4 / 5) * cos(2 * pi * n / 5)
                          + (4 / 5) * cos(4 * pi * n / 5)
                          + (8 / 7) * cos(2 * pi * n / 7)
                          + (8 / 7) * cos(4 * pi * n / 7)
                          + (8 / 7) * cos(6 * pi * n / 7))
    print(s[i])

The value of \( g(n) \) in the formula above (or equivalently the value of the variable i in the code) can be viewed as a flag in which each bit corresponds to whether \( \mathtt{Fizz}, \) \( \mathtt{Buzz} \) or \( \mathtt{Jazz} \) appears in the \( n \)th term of the sequence. All other details follow from this idea.