Code to search for numbers with large persistence of multiplication. Finds the largest known ones in under a second.

high_persistence_of_multiplication_search.py

def persistence(n):
    t = 0
    while n >= 10:
        t += 1
        n2 = 1
        for d in str(n):
            n2 *= int(d)
        n = n2
    return t


best = 5
for total in range(20000):
    for a in [False, True]:
        for b in range(total):
            if a:
                k2 = b
                k5 = 0
            else:
                k2 = 0
                k5 = b
            for k3 in range(total - b):
                k7 = total - k2 - k3 - k5
                n = 2**k2 * 3**k3 * 5**k5 * 7**k7
                s = persistence(n)
                if s >= best:
                    m = int('2'*k2 + '3'*k3 + '5'*k5 + '7'*k7)
                    print(persistence(m), m, 'from 2**{} * 3**{} * 5**{} * 7**{}'.format(k2, k3, k5, k7))
                    best = s