Problem:


Score : 400 points

Problem Statement

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.

Note

When a positive integer A divides a positive integer B, A is said to a divisor of B.
For example, 6 has four divisors: 1, 2, 3 and 6.

Constraints

1 \leq N \leq 100

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the number of the Shichi-Go numbers that are divisors of N!.

Sample Input 1

9

Sample Output 1

0

There are no Shichi-Go numbers among the divisors of 9! = 1 \times 2 \times ... \times 9 = 362880.

Sample Input 2

10

Sample Output 2

1

There is one Shichi-Go number among the divisors of 10! = 3628800: 32400.

Sample Input 3

100

Sample Output 3

543



Sample input:
9


Sample output:
0

Submission status: Accepted

Solution code:
def main(sample_file = ''):

    """ convenient functions
    # for i, a in enumerate(iterable)
    # q, mod = divmod(a, b)
    # divmod(x, y) returns the tuple (x//y, x%y)
    # Higher-order function: reduce(operator.mul, xyz_count, 1)
    # manage median(s) using two heapq https://atcoder.jp/contests/abc127/tasks/abc127_f
    """

    """convenient decorator
    # @functools.lru_cache():
    # to facilitate use of recursive function
        # ex:
        # from functools import lru_cache
        # import sys
        # sys.setrecursionlimit(10**9)
        # @lru_cache(maxsize=None)
        # def fib(n):
        #     if n < 2:
        #         return n
        #     return fib(n-1) + fib(n-2)
        # print(fib(1000))
    """
    
    # import numpy as np
    import sys
    sys.setrecursionlimit(10**7)
    from itertools import accumulate, combinations, permutations, product # https://docs.python.org/ja/3/library/itertools.html
    # accumulate() returns iterator! to get list: list(accumulate())
    from math import factorial, ceil, floor, sqrt
    def factorize(n):
        """return the factors of the Arg and count of each factor
        
        Args:
            n (long): number to be resolved into factors
        
        Returns:
            list of tuples: factorize(220) returns [(2, 2), (5, 1), (11, 1)]
        """
        fct = []  # prime factor
        b, e = 2, 0  # base, exponent
        while b * b <= n:
            while n % b == 0:
                n = n // b
                e = e + 1
            if e > 0:
                fct.append((b, e))
            b, e = b + 1, 0
        if n > 1:
            fct.append((n, 1))
        return fct
    def combinations_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        # TODO: How should I do when n - r is negative?
        if n < 0 or r < 0:
            raise Exception('combinations_count(n, r) not defined when n or r is negative')
        if n - r < r: r = n - r
        if r < 0: return 0
        if r == 0: return 1
        if r == 1: return n
        numerator = [n - r + k + 1 for k in range(r)]
        denominator = [k + 1 for k in range(r)]
        for p in range(2,r+1):
            pivot = denominator[p - 1]
            if pivot > 1:
                offset = (n - r) % p
                for k in range(p-1,r,p):
                    numerator[k - offset] /= pivot
                    denominator[k] /= pivot
        result = 1
        for k in range(r):
            if numerator[k] > 1:
                result *= int(numerator[k])
        return result
    def combinations_with_replacement_count(n, r):
        """Return the number of selecting r pieces of items from n kinds of items allowing individual elements to be repeated more than once.
        
        Args:
            n (long): number
            r (long): number
        
        Raises:
            Exception: not defined when n or r is negative
        
        Returns:
            long: number
        """
        if n < 0 or r < 0:
            raise Exception('combinations_with_replacement_count(n, r) not defined when n or r is negative')
        elif n == 0:
            return 1
        else:
            return combinations_count(n + r - 1, r)
    from bisect import bisect_left, bisect_right
    from collections import deque, Counter, defaultdict # https://docs.python.org/ja/3/library/collections.html#collections.deque
    from heapq import heapify, heappop, heappush, heappushpop, heapreplace,nlargest,nsmallest # https://docs.python.org/ja/3/library/heapq.html
    from copy import deepcopy, copy # https://docs.python.org/ja/3/library/copy.html
    import operator
    from operator import itemgetter #sort 
    # ex1: List.sort(key=itemgetter(1))
    # ex2: sorted(tuples, key=itemgetter(1,2))
    from functools import reduce, lru_cache
    def chmin(x, y):
        """change minimum
        if x > y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current minimum value
            y (long): potential minimum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x > y:
            x = y
            return (x, True)
        else:
            return (x, False)
    def chmax(x, y):
        """change maximum
        if x < y, x = y and return (x, True).
        convenient when solving problems of dp[i]
        
        Args:
            x (long): current maximum value
            y (long): potential maximum value
        
        Returns:
            (x, bool): (x, True) when updated, else (x, False)
        """
        if x < y:
            x = y
            return (x, True)
        else:
            return (x, False)

    from fractions import gcd # Deprecated since version 3.5: Use math.gcd() instead.
    def gcds(numbers):
        return reduce(gcd, numbers)
    def lcm(x, y):
        return (x * y) // gcd(x, y)
    def lcms(numbers):
        return reduce(lcm, numbers, 1)
    def make_divisors(n, reversed=False):
        """create list of divisors
        
        Args:
            number (int): number from which list of divisors is created
            reversed (bool, optional): ascending order if False. Defaults to False.
        
        Returns:
            list: list of divisors
        """
        divisors = set()
        for i in range(1, int(n**0.5)+1):
            if n % i == 0:
                divisors.add(i)
                divisors.add(n//i)
        return sorted(list(divisors),reverse=reversed)
    
    # first create factorial_list
    # fac_list = mod_factorial_list(n)
    INF = 10 ** 18
    MOD = 10 ** 9 + 7
    modpow = lambda a, n, p = MOD: pow(a, n, p) # Recursive function in python is slow!
    def modinv(a, p = MOD):
        # evaluate reciprocal using Fermat's little theorem:
        # a**(p-1) is identical to 1 (mod p) when a and p is coprime
        return modpow(a, p-2, p)
    def modinv_list(n, p = MOD):
        if n <= 1:
            return [0,1][:n+1]
        else:
            inv_t = [0,1]
            for i in range(2, n+1):
                inv_t += [inv_t[p % i] * (p - int(p / i)) % p]
            return inv_t
    def modfactorial_list(n, p = MOD):
        if n == 0:
            return [1]
        else:
            l = [0] * (n+1)
            tmp = 1
            for i in range(1, n+1):
                tmp = tmp * i % p
                l[i] = tmp
            return l
    def modcomb(n, k, fac_list = [], p = MOD):
        # fac_list = modfactorial_list(100)
        # print(modcomb(100, 5, modfactorial_list(100)))
        from math import factorial
        if n < 0 or k < 0 or n < k: return 0
        if n == 0 or k == 0: return 1
        if len(fac_list) <= n:
            a = factorial(n) % p
            b = factorial(k) % p
            c = factorial(n-k) % p
        else:
            a = fac_list[n]
            b = fac_list[k]
            c = fac_list[n-k]
        return (a * modpow(b, p-2, p) * modpow(c, p-2, p)) % p
    def modadd(a, b, p = MOD):
        return (a + b) % MOD
    def modsub(a, b, p = MOD):
        return (a - b) % p
    def modmul(a, b, p = MOD):
        return ((a % p) * (b % p)) % p
    def moddiv(a, b, p = MOD):
        return modmul(a, modpow(b, p-2, p))

    class UnionFindTree:
        """union find tree class
        TODO: fix this description...
        how to use (example):
        >>  uf = UnionFindTree(N) 
        >>  if uf.find_root(a) == uf.find_root(b):
        >>      do something
        >>  else:
        >>      do something
        >>      uf.unite(a, b)
        """
        def __init__(self, N):
            self.root = [-1] * (N+1)
            self.rank   = [0]  * (N+1)
            self.connected_num = [1] * (N+1)
    
        def find_root(self,x):
            root = self.root
            while root[x] != -1:
                x = root[x]
            return x
    
        def unite(self,x,y):
            root = self.root
            rank = self.rank
            connected_num = self.connected_num
            find_root = self.find_root
    
            rx = find_root(x)
            ry = find_root(y)
            if rx != ry:
                if rank[rx] < rank[ry]:
                    root[rx] = ry
                    rx,ry = ry,rx
                else:
                    if rank[rx] == rank[ry]:
                        rank[rx] += 1
                    root[ry] = rx       
                connected_num[rx] += connected_num[ry]

    # Graph: https://en.wikipedia.org/wiki/Directed_graph
    # Bellman-Ford: O(|V||E|). Use this if there exists an edge with negative length in the graph
    # After N steps, the shortest path has converded if there doesn't exist an cycle of edges with negative
    # Watch out: d[N] == d[2*N] doesn't necessarily mean the graph doesn't have negative cycle
    # ref: https://www.youtube.com/watch?v=1Z6ofKN03_Y

    def BellmanFord(N, M, ABC, vertex_start, vertex_end, value_if_inf = -1, find_shortest = False):
        """to calculate furthest or shortest length between vertex_start and vertex_end using BellmanFord algorithm
        
        Args:
            N (int): number of vertices
            M (int): number of edges
            ABC (list): [(ai, bi, ci) for _ in range(N)] where i-th edge is directed from vertex ai to vertex bi and the length is ci 
            vertex_start (int): start vertex. usually use 0.
            vertex_end (int): end vertex. usually use N-1.
            value_if_inf (int or string as you like, optional): value you want when the furthest (or shortest) distance is infinite (or -infinite). Defaults to -1.
            find_shortest (bool, optional): choose False to find furthest path. Defaults to False.
        
        Returns:
            int or string: normally int (but can be str if you set value_if_inf to str)
            
        Example:
                N, M, P = R()
                ABC = [R() for _ in range(M)]
                ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
                print(BellmanFord(N, M, ABC, 0, N-1, value_if_inf = 'inf'))

        """

        def make_reachable_list(N, M, ABC, vertex_start, vertex_end):
            reachable_to_direct = defaultdict(list)
            reachable_from_direct = defaultdict(list)
            reachable_from_start = [False] * N
            reachable_to_end = [False] * N
            reachable_from_start[vertex_start] = True
            reachable_to_end[vertex_end] = True
            reachable_from_both_sides = [False] * N
            dfs_from_start = []
            dfs_to_end = []
            for a, b, c in ABC:
                reachable_to_direct[a].append(b)
                reachable_from_direct[b].append(a)
                if a == vertex_start:
                    dfs_from_start.append(b)
                    reachable_from_start[b] = True
                if b == vertex_end:
                    dfs_to_end.append(a)
                    reachable_to_end[a] = True
            while dfs_from_start:
                v = dfs_from_start.pop()
                for i in reachable_to_direct[v]:
                    if not reachable_from_start[i]:
                        reachable_from_start[i] = True
                        dfs_from_start.append(i)
            while dfs_to_end:
                v = dfs_to_end.pop()
                for i in reachable_from_direct[v]:
                    if not reachable_to_end[i]:
                        reachable_to_end[i] = True
                        dfs_to_end.append(i)
            for i in range(N):
                reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
            return reachable_from_both_sides
        
        reachable_from_both_sides = make_reachable_list(N, M, ABC, vertex_start, vertex_end)
        
        if find_shortest:
            dist = [INF for i in range(N)]
        else:
            dist = [-INF for i in range(N)]
        
        dist[vertex_start] = 0
        for i in range(N):
            updated = False
            for a, b, c in ABC:
                if not reachable_from_both_sides[a]:
                    continue
                elif find_shortest:
                    update_condition = dist[a] + c < dist[b]
                else:
                    update_condition = dist[a] + c > dist[b]
                if dist[a] != INF and update_condition:
                    dist[b] = dist[a] + c
                    updated = True
                    if i == N-1:
                        return value_if_inf
            if not updated:
                break
        return dist[vertex_end]

    """ initialize variables and set inputs
    # initialize variables
        # to initialize list, use [0] * n
        # to initialize two dimentional array, use [[0] * N for _ in range(N)]
    # set inputs
        # put inputs between specific values (almost as quickly)
        # ex) S = [-INF] + [int(r()) for _ in range(A)] + [INF]
        # open(0).read() is sometimes useful:
        # ex) n, m, *x = map(int, open(0).read().split())
        #     min(x[::2]) - max(x[1::2])
        # ex2) *x, = map(int, open(0).read().split())
        #     don't forget to add comma after *x if only one variable is used
    # preprocessing
        # transpose = [x for x in zip(*data)]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [(1, 4, 7), (2, 5, 8), (3, 6, 9)]
        # flat = [flatten for inner in data for flatten in inner]
        # ex) [[1, 2, 3], [4, 5, 6], [7, 8, 9]] => [1, 2, 3, 4, 5, 6, 7, 8, 9]
    # calculate and output
        # output pattern
        # ex1) print(*l) => when l = [2, 5, 6], printed 2 5 6
    """

    # functions to read input
    r = lambda: sys.stdin.readline().strip()
    r_int = lambda: int(r())
    R = lambda: list(map(int, r().split()))
    Rmap = lambda: map(int, r().split())
    Rfloat = lambda: list(map(float, r().split()))
    Rtuple = lambda: tuple(map(int, r().split()))

    """ how to treat input
    # single int: int(r())
    # single string: r()
    # single float: float(r())
    # line int: R()
    # line string: r().split()
    # line (str, int, int): [j if i == 0 else int(j) for i, j in enumerate(r().split())]
    # lines int: [R() for _ in range(n)]
    """

    # for test
    if sample_file:
        sys.stdin = open(sample_file)
        
    # ----------------------------------
    # main

    # N = int(r())
    # e = [0] * (N+1)
    # for i in range(2, N+1):
    #     cur = i
    #     for j in range(2, i+1):
    #         while cur % j == 0:
    #             e[j] += 1
    #             cur //= j
    # def num(m): # e の要素のうち m-1 以上のものの個数
    #     return len(list(filter(lambda x: x >= m-1, e)))
    # print(num(75) + num(25) * (num(3) - 1) + num(15) * (num(5) - 1) + num(5) * (num(5) - 1) * (num(3) - 2) // 2)
    # print(num(75), num(25), num(15), num(5), num(3))

    N = int(r())
    l = [0] * 5
    prime = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]
    for j in prime:
        k, cnt = 1, 0
        while j**k < 100:
            k += 1
        k -= 1
        for i in range(2, N+1):
            k_ = k
            while k_ >= 1:
                if i%(j**k_) == 0:
                    cnt += k_
                    break
                k_ -= 1
        if cnt >= 74:
            l[4] += 1
        elif cnt >= 24:
            l[3] += 1
        elif cnt >= 14:
            l[2] += 1
        elif cnt >= 4:
            l[1] += 1
        elif cnt >= 2:
            l[0] += 1
    cmb = combinations_count
    s, t, u, v, w = l[0], l[1], l[2], l[3], l[4]
    ans = s * cmb(t+u+v+w, 2) \
        + cmb(t+u+v+w, 3) * 3 \
        + cmb(s+t+u,1)*cmb(v+w, 1) \
        + cmb(v+w, 2) * 2 \
        + t * cmb(u+v+w, 1) \
        + cmb(u+v+w, 2) * 2 \
        + w
    print(ans)

    # N, M, P = R()
    # ABC = [R() for _ in range(M)]
    # ABC = [(a-1, b-1, c-P) for a, b, c in ABC]
    
    # vertex_start, vertex_end = 0, N-1
    # reachable_from_start = [False] * N
    # reachable_to_end = [False] * N
    # reachable_from_start[vertex_start] = True
    # reachable_to_end[vertex_end] = True
    # reachable_from_both_sides = [False] * N

    # dfs_from_start = []
    # dfs_to_end = []
    # reachable_to_direct = defaultdict(list)
    # reachable_from_direct = defaultdict(list)

    # for a, b, c in ABC:
    #     reachable_to_direct[a].append(b)
    #     reachable_from_direct[b].append(a)
    #     if a == vertex_start:
    #         dfs_from_start.append(b)
    #         reachable_from_start[b] = True
    #     if b == vertex_end:
    #         dfs_to_end.append(a)
    #         reachable_to_end[a] = True
    
    # while dfs_from_start:
    #     v = dfs_from_start.pop()
    #     for i in reachable_to_direct[v]:
    #         if not reachable_from_start[i]:
    #             reachable_from_start[i] = True
    #             dfs_from_start.append(i)

    # while dfs_to_end:
    #     v = dfs_to_end.pop()
    #     for i in reachable_from_direct[v]:
    #         if not reachable_to_end[i]:
    #             reachable_to_end[i] = True
    #             dfs_to_end.append(i)
    
    # for i in range(N):
    #     reachable_from_both_sides[i] = reachable_from_start[i] and reachable_to_end[i]
    
    # dist = [-INF for i in range(N)]
    # dist[vertex_start] = 0
    # for i in range(N):
    #     updated = False
    #     for a, b, c in ABC:
    #         if not reachable_from_both_sides[a]:
    #             continue
    #         if dist[a] != INF and dist[a] + c > dist[b]:
    #             dist[b] = dist[a] + c
    #             updated = True
    #             if i == N-1:
    #                 print(-1)
    #                 return
    #     if not updated:
    #         break
    # print(max(dist[vertex_end], 0))
                
    # to finish code, use return instead of exit()
    # end of main
    # ----------------------------------

    """memo: how to use defaultdict of list
    # initialize
    Dic = defaultdict(list)
    # append / extend
    Dic[x].append(y)
    # three methods for loop: keys(), values(), items()
    for k, v in Dic.items():
    """
    
    """memo: how to solve binary problems
    # to make binary digits text
    >>> a = 5
    >>> bin_str_a = format(a, '#06b')
    >>> print(bin_str_a)
    0b0101 # first 2 strings (='0b') indicates it is binary
    """
    
    """memo: how to solve the problem
    creating simple test/answer
    greed
    simple dp
    graph
    """

if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
N = int(input())

prime = [0] * (N + 1)
for i in range(1, N + 1) :
    while i % 2 == 0 :
        i //= 2
        prime[2] += 1
    j = 3
    while i != 1 :
        while i % j == 0 :
            i //= j
            prime[j] += 1
        j += 2

# 3 5 15 25 75
over = [0] * 5

for i in range(1, N + 1) :
    if 74 <= prime[i] :
        over[4] += 1
    elif 24 <= prime[i] :
        over[3] += 1
    elif 14 <= prime[i] :
        over[2] += 1
    elif 4 <= prime[i] :
        over[1] += 1
    elif 2 <= prime[i] :
        over[0] += 1
        
s = [sum(over[i:]) for i in range(5)]

ret = 0
# 3*5*5
ret += over[0] * s[1] * (s[1] - 1) // 2
ret += s[1] * (s[1] - 1) * (s[1] - 2) // 2
# 3*25
ret += (over[0] + over[1] + over[2]) * s[3]
ret += s[3] * (s[3] - 1)
# 5*15
ret += over[1] * s[2]
ret += s[2] * (s[2] - 1)
# 75
ret += over[4]
print(ret)

Submission status: Accepted

Solution code:
n = int(input())

p = [0] * 101

for i in range(1, n+1):
    tmp = i
    while tmp > 1:
        j = 2
        while tmp % j != 0:
            j+=1
        tmp = tmp //j
        p[j] += 1
p = [x for x in p if x > 0]

ans = 0
lp = len(p)
for i in range(lp):
    for j in range(lp):
        for k in range(j+1, lp):
            if i != j and i != k and p[i] >= 2 and p[j] >= 4 and p[k] >= 4:
                ans += 1

for i in range(lp):
    for j in range(lp):
        if i != j and p[i] >= 14 and p[j] >= 4:
            ans += 1
for i in range(lp):
    for j in range(lp):
        if i != j and p[i] >= 24 and p[j] >= 2:
            ans += 1
for i in range(lp):
    if p[i] >= 74:
        ans += 1
print(ans)

Submission status: Accepted

Solution code:
N = int(input())

prime_number = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

a = 0
_count = []
ans = 0

for g in range(len(prime_number)):
	if prime_number[g] > N:
		break
	for i in range(1, 100):
		a += N//prime_number[g]**i
		if  prime_number[g]**i > N:
			_count.append(a)
			a=0
			break

if N!=1:
	if _count[0] >= 74:
		ans+=1

c_24 = []
c_14 = []
c_4 = []
c_2 = []
for n in range(len(_count)):
	if _count[n]>=24:
		c_24.append(_count[n])
	if _count[n]>=14:
		c_14.append(_count[n])
	if _count[n]>=4:
		c_4.append(_count[n])
	if _count[n]>=2:
		c_2.append(_count[n])

ans += len(c_14)*(len(c_4)-1)
ans += len(c_24)*(len(c_2)-1)
ans += (len(c_4)-1)*len(c_4)*(len(c_2)-2)/2

print(int(max(ans,0)))

Submission status: Accepted

Solution code:
from fractions import math

N = int(input())

def prime(n):
    factor = {}
    tmp = int(math.sqrt(n)) + 1
    for num in range(2, tmp):
        while n % num == 0:
            n //= num
            if not num in factor.keys():
                factor[num] = 1
            else:
                factor[num] += 1
    if n != 1:
        if not n in factor.keys():
            factor[n] = 1
        else:
            factor[n] += 1
    return factor

Factor = {}
for n in range(2, N+1):
    ns = prime(n)
    for k, v in ns.items():
        if not k in Factor.keys():
            Factor[k] = v
        else:
            Factor[k] += v

c74 = 0
c24 = 0
c14 = 0
c4 = 0
c2 = 0
for num, co in Factor.items():
    if co >= 74:
        c74 += 1
    if co >= 24:
        c24 += 1
    if co >= 14:
        c14 += 1
    if co >= 4:
        c4 += 1
    if co >= 2:
        c2 += 1

ans = c74 + c24*(c2-1) + c14*(c4-1) + c4*(c4-1)*(c2-2)//2
print(ans)

Submission status: Accepted

Solution code:
def main():
    n = int(input())
    l_prm = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
    l_prm = [i for i in l_prm if i<=n/2]
    l_new = []
    for i in l_prm:
        a = n
        counter = 0
        b = a
        c=1
        while b!=0:
            b = a//i**c
            counter += b
            c += 1
        l_new.append(counter)
    ans=0
    def num(j,lst):
        return(len([i for i in lst if i>=j]))
    ans += num(4,l_new) * (num(4,l_new)-1)/2 *(num(2,l_new)-2)
    ans += num(14,l_new)*(num(4,l_new)-1)
    ans += num(24,l_new)*(num(2,l_new)-1)
    ans += num(74,l_new)
    print(int(ans))
if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:
N = int(input())
#1~50の素数を見つける
a=[i for i in range(2,51)]
sosu = [2]
while(True):
    a = [i for i in a if i%sosu[-1]!=0]
    if a == []:
        break
    sosu.append(a[0])
nsosu = [0]*len(sosu)
for n in range(2,N+1):
    for ind,s in enumerate(sosu):
        if n%s == 0:
            count = 0
            while(n%s == 0):
                count += 1
                n = n/s
            nsosu[ind] += count
        elif int(n/s) == 0:
            if n != 1:
                nsosu.append(1)
            break
        if ind == len(sosu)-1:
            if n != 1:
                nsosu.append(1)
nsosu = [i for i in nsosu if i !=0]
#8*5*2*2=160個の約数
#75 = (1,75) (3,25),(5,15),(3,5,5)
#(3,25)を探す
count = 0
count5 = 0
for i,ns in enumerate(nsosu):
    if ns >= 24:
        for j,nso in enumerate(nsosu):
            if i != j:
                if nso>=2:
                    count += 1
                    
    #(5,15)を探す
    if ns >= 14:
        for j,nso in enumerate(nsosu):
            if i != j:
                if nso >= 4:
                    count += 1
    #(3,5,5)
    if ns >= 4:
        for j,nso in enumerate(nsosu):
            if i != j:
                if nso >= 4:
                    for k,nsos in enumerate(nsosu):
                        if i != k and j != k:
                            if nsos >= 2:
                                count5 += 1
    #(1,75)
    if ns >= 74:
        count += 1
count += int(count5/2)                  
print(count)

Submission status: Accepted

Solution code:
#%%
N =int(input())

#import sympy

l = [0] * (N+1)

#for i in range(1,N+1):
#    a = sympy.factorint(i)
#    for k, v in a.items():
#        l[k] += v


def factorize(n):
    fct = []  # prime factor
    b, e = 2, 0  # base, exponent
    while b * b <= n:
        while n % b == 0:
            n = n // b
            e = e + 1
        if e > 0:
            fct.append((b, e))
        b, e = b + 1, 0
    if n > 1:
        fct.append((n, 1))
    return fct

for j in range(1, N+1):
    a = factorize(j)
#    print(a)
    for i in range(len(a)):
        l[a[i][0]] += a[i][1]


#print(l)

l_2 = []
l_4 = []
l_14 = []
l_24 = []
l_74 = []

for i in range(len(l)):
    if l[i]>=2:
        l_2.append(i)
    if l[i]>=4:
        l_4.append(i)
    if l[i]>=14:
        l_14.append(i)
    if l[i]>=24:
        l_24.append(i)
    if l[i]>=74:
        l_74.append(i)

#print(l_2)
#print(l_4)
#print(l_14)
#print(l_24)
#print(l_74)

count_2_4_4 = 0
count_4_14 = 0
count_2_24 = 0
count_74 = 0

for i in range(len(l_2)):
    for j in range(len(l_4)):
        for k in range(j,len(l_4)):
            if l_2[i] != l_4[j] and l_2[i] != l_4[k] and l_4[j] != l_4[k]:
                count_2_4_4 += 1

for i in range(len(l_4)):
    for j in range(len(l_14)):
        if l_4[i] != l_14[j] :
            count_4_14 += 1

for i in range(len(l_2)):
    for j in range(len(l_24)):
        if l_2[i] != l_24[j]:
            count_2_24 += 1

count_74 += len(l_74)


#print(count_2_4_4)
#print(count_4_14)
#print(count_2_24)
#print(count_74)

ans = count_2_4_4 + count_4_14 + count_2_24 + count_74
print(ans)

Submission status: Accepted

Solution code:
import itertools

n = int(input())

def prime(n):
    number = [i+2 for i in range(n-1)]
    while len(number) > 0:
        yield min(number)
        number = [x for x in number if x % min(number) != 0]
p = {}
for i in prime(n):
    p.setdefault(i, 0)
for i in range(n):
    test = i + 1
    for j in p:
        while test % j == 0:
            p[j] += 1
            test //= j
c = [[75], [3, 25], [5, 15], [3, 5, 5]]
count = 0
for i in c:
    for j in itertools.permutations(p.keys(), len(i)):
        flag = True
        for k in range(len(i)):
            if p[j[k]] <= i[k] - 2:
                flag = False
                break
        if flag == True:
            if i == [3, 5, 5]:
                count += 0.5
            else:
                count += 1
print(int(count))

Submission status: Accepted

Solution code:
#平方数でなければ約数は奇数とはならない
N= int(input())
e = [0] * (N+1)
for i in range(2, N+1):
  cur = i
  for j in range(2, i+1):
    while cur % j == 0:
      e[j] += 1
      cur //= j

def num(m): # e の要素のうち m-1 以上のものの個数
  return len(list(filter(lambda x: x >= m-1, e)))

ans = num(75) + num(25)*(num(3)-1) + num(15)*(num(5)-1) + num(5)*(num(5)-1)*(num(3)-2)//2

print(ans)

Submission status: Accepted

Solution code:
from math import factorial
from itertools import permutations

N = int(input())

M = factorial(N)


primes = [True for _ in range(N + 1)]
primes[0] = primes[1] = False
for i in range(2, N + 1):
    if primes[i]:
        for j in range(2 * i, N + 1, i):
            primes[j] = False


# m >= k^i を満たす最大のiを返す
def f(m, k):
    return (
        1 + f(m // k, k) if m % k == 0 else
        0
    )


# 素因数分解した結果に着目して、七五数になるパターンごとに数え上げ
C = [f(M, k) for k in range(2, N + 1) if primes[k]]
ans = (
    sum(c >= 74 for (c,) in permutations(C, 1)) +
    sum(c1 >= 24 and c2 >= 2 for (c1, c2) in permutations(C, 2)) +
    sum(c1 >= 14 and c2 >= 4 for (c1, c2) in permutations(C, 2)) +
    sum(
        c1 >= 4 and c2 >= 4 and c3 >= 2
        for (c1, c2, c3) in permutations(C, 3)
    ) // 2
)

print(ans)

Submission status: Accepted

Solution code:
N = int(input())

pf = [0] * (N + 1)
for n in range(2, N + 1):
    cur = n
    for i in range(2, n + 1):
        while cur % i == 0:
            pf[i] += 1
            cur //= i

def num(n):
    return len(list(filter(lambda x: x >= n - 1, pf)))

print(num(75) + num(25) * (num(3) - 1) + num(15) * (num(5) - 1) \
          + num(5) * (num(5) - 1) * (num(3) - 2) // 2)

Submission status: Accepted

Solution code:
#!/usr/bin/env python3
#ABC114 D

import sys
import math
import bisect
from heapq import heappush, heappop
from collections import defaultdict
from itertools import accumulate
from collections import Counter
from collections import deque
from operator import itemgetter
from itertools import permutations
mod = 10**9 + 7

n = int(input())
e = [0]*(n+1)
for i in range(2,n+1):
    cur = i
    for j in range(2,i+1):
        while cur % j == 0:
            e[j] += 1
            cur //= j

def num(m):
    return len(list(filter(lambda x: x >= m-1, e)))

ans = num(75) + num(25) * (num(3) - 1) + num(15) * (num(5) - 1) + num(5) * (num(5) - 1) * (num(3) - 2) // 2
print(ans)

Submission status: Accepted

Solution code:
N = int(input())

def primeFact(n) :
    primeList = {}
    for m in range(2, n + 1) :
        k = m
        for i in range(2, k + 1) :
            if k % i == 0 :
                if not i in primeList :
                    primeList[i] = 0
                while k % i == 0 :
                    primeList[i] += 1
                    k //= i
    return primeList

def dictFilter (filter, dic) :
    return { key : val for key, val in dic.items() if filter(key, val)}

primeN = primeFact(N)
prime75 = len(dictFilter(lambda key, val : val + 1 >= 75, primeN))
prime25 = len(dictFilter(lambda key, val : val + 1 >= 25, primeN))
prime15 = len(dictFilter(lambda key, val : val + 1 >= 15, primeN))
prime5 = len(dictFilter(lambda key, val : val + 1 >= 5, primeN))
prime3 = len(dictFilter(lambda key, val : val + 1 >= 3, primeN))

ans = 0
ans += prime75
ans += prime25 * (prime3 - 1)
ans += prime15 * (prime5 - 1)
ans += prime5 * (prime5 - 1) * (prime3 - 2) // 2

print(ans)