結果
| 問題 |
No.1039 Project Euler でやれ
|
| ユーザー |
 qwewe
|
| 提出日時 | 2025-05-14 13:01:25 |
| 言語 | PyPy3 (7.3.15) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 9,227 bytes |
| コンパイル時間 | 337 ms |
| コンパイル使用メモリ | 81,896 KB |
| 実行使用メモリ | 86,360 KB |
| 最終ジャッジ日時 | 2025-05-14 13:03:40 |
| 合計ジャッジ時間 | 2,185 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge4 |
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| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 15 WA * 3 |
ソースコード
import sys
import itertools
# Set a higher recursion depth limit for potentially deep recursion in partition generation
# Although max exponent K is relatively small (~20 for M < 10^6), the structure of recursion might need it.
sys.setrecursionlimit(2000)
# Define the modulus P = 10^9 + 7
MOD = 10**9 + 7
# Function for modular exponentiation (base^power % MOD)
# Uses binary exponentiation (also known as exponentiation by squaring) for efficiency
def fast_pow(base, power):
result = 1
while power > 0:
# If power is odd, multiply result with base
if power % 2 == 1:
result = (result * base) % MOD
# Square the base and halve the power
base = (base * base) % MOD
power //= 2
return result
# Function for modular inverse using Fermat's Little Theorem
# Calculates a^(MOD-2) % MOD, which is the modular inverse of a modulo MOD (since MOD is prime)
def inverse(a):
# Check if a is 0 modulo MOD. If so, its inverse doesn't exist.
# The size of an automorphism group |Aut(G)| is always >= 1 for |G| >= 1, so a should not be 0.
if a == 0:
raise ValueError("Modular inverse of 0 requested")
return fast_pow(a, MOD - 2)
# Precompute factorials modulo MOD up to the maximum possible value of M
# M < 10^6, so we compute up to 10^6.
MAX_M_limit = 10**6
fact = [1] * (MAX_M_limit + 1)
for i in range(1, MAX_M_limit + 1):
fact[i] = (fact[i - 1] * i) % MOD
# Function to calculate the size of the automorphism group of an abelian p-group
# G = Z_{p^{e_1}} x ... x Z_{p^{e_n}}
# The input 'partition' is a list of exponents [e1, e2, ..., en]
# The formula used requires the partition structure represented as:
# G = (Z_{p^{lambda_1}})^{m_1} x ... x (Z_{p^{lambda_k}})^{m_k} where lambda_1 > lambda_2 > ... > lambda_k > 0
# This function handles converting the input partition list to this structure internally.
def calculate_aut_p_group_size(p, partition):
# Count occurrences of each exponent value in the input partition
counts = {}
for x in partition:
if x > 0: # Partition parts must be positive integers
counts[x] = counts.get(x, 0) + 1
# If the partition is empty (this corresponds to k=0 exponent, for M=1),
# the group is the trivial group {e}, and |Aut({e})| = 1.
if not counts:
return 1
# Get the distinct exponents (lambda_i) and sort them in descending order
distinct_exponents = sorted(counts.keys(), reverse=True)
lambdas = distinct_exponents
# Get the multiplicities (m_i) corresponding to the sorted distinct exponents
ms = [counts[lam] for lam in lambdas]
k = len(lambdas) # Number of distinct exponent types
# The formula for |Aut(G)| has three main product terms. Calculate them modularly.
# Term 1: Product related to GL(m_i, F_p) factors
term1 = 1
for i in range(k): # Loop over distinct exponent types lambda_i
m_i = ms[i]
# Calculate p^{m_i} mod MOD
p_pow_m_i = fast_pow(p, m_i)
current_prod = 1
# Inner product runs from j=0 to m_i-1
for j in range(m_i):
# Calculate p^j mod MOD
p_pow_j = fast_pow(p, j)
# Calculate (p^{m_i} - p^j) mod MOD. Add MOD before taking modulo to handle potential negative result.
val = (p_pow_m_i - p_pow_j + MOD) % MOD
current_prod = (current_prod * val) % MOD
term1 = (term1 * current_prod) % MOD
# Term 2: Product over pairs of distinct exponents (1 <= i < j <= k)
term2 = 1
for i in range(k):
for j in range(i + 1, k): # j > i
lambda_j = lambdas[j]
m_i = ms[i]
m_j = ms[j]
# Calculate (p^{lambda_j})^(m_i * m_j) mod MOD
factor1 = fast_pow(p, lambda_j)
factor1_pow = m_i * m_j # Exponent is relatively small, direct computation is fine
term2_part1 = fast_pow(factor1, factor1_pow)
# Calculate (p^{m_j})^(m_i * m_j) mod MOD
factor2 = fast_pow(p, m_j)
factor2_pow = m_i * m_j
term2_part2 = fast_pow(factor2, factor2_pow)
# Accumulate product terms modulo MOD
term2 = (term2 * term2_part1) % MOD
term2 = (term2 * term2_part2) % MOD
# Term 3: Product related to powers p^{lambda_i - 1}
term3 = 1
for i in range(k): # Loop over distinct exponent types lambda_i
lambda_i = lambdas[i]
m_i = ms[i]
# Base is p^{lambda_i - 1}
factor = fast_pow(p, lambda_i - 1)
# Exponent is m_i^2
exponent = m_i * m_i
# Calculate (p^{lambda_i - 1})^(m_i^2) mod MOD
term3_part = fast_pow(factor, exponent)
# Accumulate product term modulo MOD
term3 = (term3 * term3_part) % MOD
# Combine all three terms to get the final size of the automorphism group
total_aut_size = (term1 * term2) % MOD
total_aut_size = (total_aut_size * term3) % MOD
return total_aut_size
# Function for prime factorization of an integer n
def prime_factorize(n):
factors = {}
d = 2
# Use trial division up to sqrt(n)
while d * d <= n:
while n % d == 0:
factors[d] = factors.get(d, 0) + 1
n //= d
d += 1
# If n is still greater than 1 after the loop, it must be a prime factor itself
if n > 1:
factors[n] = factors.get(n, 0) + 1
return factors
# Function to generate integer partitions of k with memoization
# Generates partitions as lists of integers in non-decreasing order, e.g., [1, 1, 2] for k=4.
memo_partitions = {}
def generate_partitions_memo(k):
# Use k as the key for memoization
state = k
if state in memo_partitions:
return memo_partitions[state]
# Base case: Partition of 0 is represented by a list containing an empty list
if k == 0:
return [[]]
partitions = []
# 'i' represents the smallest part in the partition being constructed
for i in range(1, k + 1):
# Recursively find partitions for the remaining sum k-i
results_k_minus_i = generate_partitions_memo(k - i)
for result in results_k_minus_i:
# Ensure non-decreasing order:
# The current part 'i' must be less than or equal to the smallest part in the 'result' list (if 'result' is not empty)
if not result or result[0] >= i:
# If the condition is met, prepend 'i' to the sub-partition 'result'
partitions.append([i] + result)
# Store the generated partitions in the memoization table and return them
memo_partitions[state] = partitions
return partitions
# Main program logic
# Read input M
M = int(sys.stdin.readline())
# Handle the trivial case M=1. The only group is the trivial group on {0}. There is 1 such structure.
if M == 1:
print(1)
sys.exit() # Exit after printing the result
# Compute the prime factorization of M. Example: M=12 -> {2: 2, 3: 1}
prime_factors = prime_factorize(M)
# Get list of prime factors p_i
primes = list(prime_factors.keys())
# Get list of corresponding exponents k_i
exponents = [prime_factors[p] for p in primes]
# Generate all partitions for each exponent k_i. Store these lists of partitions.
partitions_for_exponents = []
for k_i in exponents:
partitions_for_exponents.append(generate_partitions_memo(k_i))
# Calculate M! mod MOD using the precomputed table
M_factorial = fact[M]
# Initialize the total sum of terms M! / |Aut(G)|
total_sum = 0
# An abelian group G of order M is isomorphic to a direct product of its Sylow p-subgroups G_p.
# The structure of each G_p is determined by a partition of the exponent k_i of p in M.
# We need to iterate through all combinations of partitions, one for each prime factor p_i.
# `itertools.product` generates the Cartesian product of the lists of partitions.
list_of_partition_lists = partitions_for_exponents
combinations = list(itertools.product(*list_of_partition_lists))
# Each 'combo' is a tuple of partitions, e.g., (partition_for_p1, partition_for_p2, ...).
# This tuple represents one isomorphism class of an abelian group G of order M.
for combo in combinations:
# Calculate the size of the automorphism group |Aut(G)| for this isomorphism class.
# |Aut(G)| = product of |Aut(G_p)| for each Sylow p-subgroup G_p.
current_aut_size = 1
for i in range(len(primes)):
p = primes[i] # The prime factor
partition = combo[i] # The partition defining the structure of G_p
# Calculate |Aut(G_p)| using the dedicated function
aut_p_group_size = calculate_aut_p_group_size(p, partition)
# Multiply into the total |Aut(G)| size, modulo MOD
current_aut_size = (current_aut_size * aut_p_group_size) % MOD
# Calculate the term M! / |Aut(G)| modulo MOD.
# This is equivalent to M! * (|Aut(G)|)^(-1) mod MOD.
inv_aut_size = inverse(current_aut_size)
term = (M_factorial * inv_aut_size) % MOD
# Add this term to the total sum, modulo MOD
total_sum = (total_sum + term) % MOD
# Print the final total sum modulo MOD
print(total_sum)