Problem: You are given an integer N. Among the divisors of N! (= 1 \times 2 \times ... \times N), how many _Shichi-Go numbers_ (literally "Seven-Five numbers") are there?  Here, a Shichi-Go number is a positive integer that has exactly 75 divisors.

Submission attempts (in order):
#!/usr/bin/env python3

import sys
import math
# from string import ascii_lowercase, ascii_upper_case, ascii_letters, digits, hexdigits
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from operator import itemgetter              # itemgetter(1), itemgetter('key')

# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product, permutations  # product(iter, repeat=n), permutations(iter[,r])
# from itertools import combinations, combinations_with_replacement
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict (e.g. list is not allowed)
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference
# from fractions import gcd                    # for Python 3.4 (previous contest @AtCoder)


def main():
    mod = 1000000007                # 10^9+7
    inf = float('inf')              # sys.float_info.max = 1.79...e+308
    # inf = 2 ** 64 - 1               # (for fast JIT compile in PyPy) 1.84...e+19
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():    return int(input())
    def mi():    return map(int, input().split())
    def mi_0():  return map(lambda x: int(x)-1, input().split())
    def lmi():   return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():    return list(input())
    
    
    class Eratos:
        def __init__(self, num):
            
            assert(num >= 1)
            self.table_max = num
            
            self.table = [False if i == 0 or i == 1 else True for i in range(num+1)]
            for i in range(2, int(math.sqrt(num)) + 1):
                if self.table[i]:
                    for j in range(i ** 2, num + 1, i):    
                        self.table[j] = False
        
        def is_prime(self, num):
            
            assert(num >= 1)
            if num > self.table_max:
                raise ValueError('Eratos.is_prime(): exceed table_max({}). got {}'.format(self.table_max, num))
            return self.table[num]
        
        def prime_factorize(self, num):
            
            assert(num >= 1)
            if int(math.sqrt(num)) > self.table_max:
                raise ValueError('Eratos.prime_factorize(): exceed prime table size. got {}'.format(num))
            
            factorized_dict = dict()
            candidate_prime_numbers = [i for i in range(2, int(math.sqrt(num)) + 1) if self.is_prime(i)]
            
            
            for p in candidate_prime_numbers:
                if num == 1:    
                    break
                if num % p == 0:
                    cnt = 0
                    while num % p == 0:
                        num //= p
                        cnt += 1
                    factorized_dict[p] = cnt
            if num != 1:
                factorized_dict[num] = 1
            return factorized_dict
        
        def enum_divisor(self, num):
            
            divisor_small = []
            divisor_large = []
            for i in range(1, int(math.sqrt(num)) + 1):
                if num % i == 0:
                    divisor_small.append(i)
                    if i != num // i:
                        divisor_large.append(num // i)
            divisor_large.reverse()
            return divisor_small + divisor_large
    
    n = ii()
    e = Eratos(n)
    c = Counter()
    for i in range(2, n + 1):
        c += Counter(e.prime_factorize(i))
    print(c)
    
    gt_74_candidate = 0
    gt_24_candidate = 0
    gt_14_candidate = 0
    gt_4_candidate = 0
    gt_2_candidate = 0
    for _, v in c.items():
        if v >= 2:
            gt_2_candidate += 1
        if v >= 4:
            gt_4_candidate += 1
        if v >= 14:
            gt_14_candidate += 1
        if v >= 24:
            gt_24_candidate += 1
        if v >= 74:
            gt_74_candidate += 1
    
    def calc_3_5_5(a, b):
        res = a - b
        if a >= 3 and b >= 2:
            # bC2 * resC1 + bC3
            return (b * (b-1) // 2) * res + (b * (b-1) * (b-2) // 6)
        else:
            return 0
    
    def calc_5_15(a, b):
        res = a - b
        if a >= 2 and b >= 1:
            # bC1 * resC1 + bC2
            return b * res + (b * (b-1) // 2) 
        else:
            return 0
    
    def calc_3_25(a, b):
        res = a - b
        if a >= 2 and b >= 1:
            # bC1 * resC1 + bC2
            return b * res + (b * (b-1) // 2)
        else:
            return 0
    
    def calc_75(a):
        if a >= 1:
            return a
        else:
            return 0


    def calc_all_75_num(gt_2_candidate, gt_4_candidate, gt_14_candidate, gt_24_candidate, gt_74_candidate):
        ans = 0
        ans += calc_3_5_5(gt_2_candidate, gt_4_candidate)
        print(ans)
        ans += calc_5_15(gt_4_candidate, gt_14_candidate)
        print(ans)
        ans += calc_3_25(gt_2_candidate, gt_24_candidate)
        print(ans)
        ans += calc_75(gt_74_candidate)
        # print(ans)
        return ans
    
    print(calc_all_75_num(gt_2_candidate, gt_4_candidate, gt_14_candidate, gt_24_candidate, gt_74_candidate))



if __name__ == "__main__":
    main()

#!/usr/bin/env python3

import sys
import math
# from string import ascii_lowercase, ascii_upper_case, ascii_letters, digits, hexdigits
# import re                                    # re.compile(pattern) => ptn obj; p.search(s), p.match(s), p.finditer(s) => match obj; p.sub(after, s)
# from operator import itemgetter              # itemgetter(1), itemgetter('key')

# from collections import defaultdict          # subclass of dict. defaultdict(facroty)
from collections import Counter              # subclass of dict. Counter(iter): c.elements(), c.most_common(n), c.subtract(iter)
# from heapq import heapify, heappush, heappop # built-in list. heapify(L) changes list in-place to min-heap in O(n), heappush(heapL, x) and heappop(heapL) in O(lgn).
# from heapq import nlargest, nsmallest        # nlargest(n, iter[, key]) returns k-largest-list in O(n+klgn).
# from itertools import count, cycle, repeat   # count(start[,step]), cycle(iter), repeat(elm[,n])
# from itertools import groupby                # [(k, list(g)) for k, g in groupby('000112')] returns [('0',['0','0','0']), ('1',['1','1']), ('2',['2'])]
# from itertools import starmap                # starmap(pow, [[2,5], [3,2]]) returns [32, 9]
# from itertools import product, permutations  # product(iter, repeat=n), permutations(iter[,r])
# from itertools import combinations, combinations_with_replacement
# from itertools import accumulate             # accumulate(iter[, f])
# from functools import reduce                 # reduce(f, iter[, init])
# from functools import lru_cache              # @lrucache ...arguments of functions should be able to be keys of dict (e.g. list is not allowed)
# from bisect import bisect_left, bisect_right # bisect_left(a, x, lo=0, hi=len(a)) returns i such that all(val<x for val in a[lo:i]) and all(val>-=x for val in a[i:hi]).
# from copy import deepcopy                    # to copy multi-dimentional matrix without reference
# from fractions import gcd                    # for Python 3.4 (previous contest @AtCoder)


def main():
    mod = 1000000007                # 10^9+7
    inf = float('inf')              # sys.float_info.max = 1.79...e+308
    # inf = 2 ** 64 - 1               # (for fast JIT compile in PyPy) 1.84...e+19
    sys.setrecursionlimit(10**6)    # 1000 -> 1000000
    def input(): return sys.stdin.readline().rstrip()
    def ii():    return int(input())
    def mi():    return map(int, input().split())
    def mi_0():  return map(lambda x: int(x)-1, input().split())
    def lmi():   return list(map(int, input().split()))
    def lmi_0(): return list(map(lambda x: int(x)-1, input().split()))
    def li():    return list(input())
    
    
    class Eratos:
        def __init__(self, num):
            
            assert(num >= 1)
            self.table_max = num
            
            self.table = [False if i == 0 or i == 1 else True for i in range(num+1)]
            for i in range(2, int(math.sqrt(num)) + 1):
                if self.table[i]:
                    for j in range(i ** 2, num + 1, i):    
                        self.table[j] = False
        
        def is_prime(self, num):
            
            assert(num >= 1)
            if num > self.table_max:
                raise ValueError('Eratos.is_prime(): exceed table_max({}). got {}'.format(self.table_max, num))
            return self.table[num]
        
        def prime_factorize(self, num):
            
            assert(num >= 1)
            if int(math.sqrt(num)) > self.table_max:
                raise ValueError('Eratos.prime_factorize(): exceed prime table size. got {}'.format(num))
            
            factorized_dict = dict()
            candidate_prime_numbers = [i for i in range(2, int(math.sqrt(num)) + 1) if self.is_prime(i)]
            
            
            for p in candidate_prime_numbers:
                if num == 1:    
                    break
                if num % p == 0:
                    cnt = 0
                    while num % p == 0:
                        num //= p
                        cnt += 1
                    factorized_dict[p] = cnt
            if num != 1:
                factorized_dict[num] = 1
            return factorized_dict
        
        def enum_divisor(self, num):
            
            divisor_small = []
            divisor_large = []
            for i in range(1, int(math.sqrt(num)) + 1):
                if num % i == 0:
                    divisor_small.append(i)
                    if i != num // i:
                        divisor_large.append(num // i)
            divisor_large.reverse()
            return divisor_small + divisor_large
    
    n = ii()
    e = Eratos(n)
    c = Counter()
    for i in range(2, n + 1):
        c += Counter(e.prime_factorize(i))
    # print(c)
    
    gt_74_candidate = 0
    gt_24_candidate = 0
    gt_14_candidate = 0
    gt_4_candidate = 0
    gt_2_candidate = 0
    for _, v in c.items():
        if v >= 2:
            gt_2_candidate += 1
        if v >= 4:
            gt_4_candidate += 1
        if v >= 14:
            gt_14_candidate += 1
        if v >= 24:
            gt_24_candidate += 1
        if v >= 74:
            gt_74_candidate += 1
    
    def calc_3_5_5(a, b):
        if a >= 3 and b >= 2:
            # bC2 * (a - 2)
            return (b * (b-1) // 2) * (a - 2)
        else:
            return 0
    
    def calc_5_15(a, b):
        if a >= 2 and b >= 1:
            # bC1 * (a - 1)
            return b * (a - 1)
        else:
            return 0
    
    def calc_3_25(a, b):
        if a >= 2 and b >= 1:
            # bC1 * (a - 1)
            return b * (a - 1)
        else:
            return 0
    
    def calc_75(a):
        if a >= 1:
            return a
        else:
            return 0


    def calc_all_75_num(gt_2_candidate, gt_4_candidate, gt_14_candidate, gt_24_candidate, gt_74_candidate):
        ans = 0
        ans += calc_3_5_5(gt_2_candidate, gt_4_candidate)
        # print(ans)
        ans += calc_5_15(gt_4_candidate, gt_14_candidate)
        # print(ans)
        ans += calc_3_25(gt_2_candidate, gt_24_candidate)
        # print(ans)
        ans += calc_75(gt_74_candidate)
        # print(ans)
        return ans
    
    print(calc_all_75_num(gt_2_candidate, gt_4_candidate, gt_14_candidate, gt_24_candidate, gt_74_candidate))



if __name__ == "__main__":
    main()


Verdicts: Wrong Answer

Accepted