Get the minimum n and k values in order to get a value greater than or equal to a constant c.

main.py

from scipy.special import lambertw
from scipy.special import gamma
import math
import time

e = math.e


def l_func(x):
    c = 0.036534
    return math.log((x + c) / math.sqrt(2 * math.pi))


def inv_gamma(x):
    global e
    return round(l_func(x) / lambertw([l_func(x) / e]).real[0] - 0.5)


def fact(n):
    return gamma(n + 1)


def combine(n, k):
    return fact(n) / (fact(n - k) * fact(k))


def get_lim(c, t):
    while combine(t, t / 2) < c:
        t += 1
    return t


def digits(n):
    return math.log10(n) + 1


def get_t(t, c, d=2):
    prev = d
    while combine(t, t / 2) <= c:
        prev = t
        d += 1
        t = inv_gamma(c ** d + 1)
    return prev


def get_min_k(t, max_k, c):
    min_k = max_k - 1
    while combine(t, min_k) >= c:
        min_k -= 1
    return min_k + 1


if __name__ == '__main__':

    print("Write the result of the combinatorial: ", end='')
    c = int(input())

    if c > 0:
        start = time.time()
        t = inv_gamma(c ** 2 + 1)

        t = int(get_t(t, c))
        print("start at: {}".format(t))  # print where brute force start

        t = int(get_lim(c, t))
        print("Real val: {}".format(t))

        max_k = int(t / 2) if t % 2 == 0 else int((t + 1) / 2)
        min_k = get_min_k(t, max_k, c)
        found = int(combine(t, min_k))

        print("the min value of n is: {}, max k value: {}, min k value: {}.".format(t, max_k, min_k))
        print("Value found: {}".format(found))
        print("Distance between values: {}".format(found - c))
        print("Elapsed time: {}".format(time.time() - start))

    else:
        print("Indeterminate")