tests/prime_factorization/regular.test.cpp

• View this file on GitHub
• Last update: 2026-01-24 23:50:59+09:00
• Problem: https://judge.yosupo.jp/problem/factorize

Depends on

• std::abs(x) extended for my library (tools/abs.hpp)
• $2^{\left\lceil \log_2(x) \right\rceil}$ (tools/bit_ceil.hpp)
• $2^{\left\lfloor \log_2(x) \right\rfloor}$ (tools/bit_floor.hpp)
• Bit width required to represent a given integer (tools/bit_width.hpp)
• Number of trailing zeros (tools/countr_zero.hpp)
• tools/detail/int128_t_and_uint128_t.hpp
• $\left\lfloor \log_2(x) \right\rfloor$ (tools/floor_log2.hpp)
• std::gcd(m, n) extended for my library (tools/gcd.hpp)
• Check whether a given integer is $2^n$ (tools/has_single_bit.hpp)
• Combine hash values (tools/hash_combine.hpp)
• Concept for integral types including tools::int128_t and tools::uint128_t (tools/integral.hpp)
• std::is_integral extended for tools::int128_t and tools::uint128_t (tools/is_integral.hpp)
• Miller-Rabin primality test (tools/is_prime.hpp)
• std::is_signed extended for tools::int128_t and tools::uint128_t (tools/is_signed.hpp)
• std::is_unsigned extended for tools::int128_t and tools::uint128_t (tools/is_unsigned.hpp)
• std::make_signed extended for tools::int128_t and tools::uint128_t (tools/make_signed.hpp)
• std::make_unsigned extended for tools::int128_t and tools::uint128_t (tools/make_unsigned.hpp)
• Minimum non-negative reminder (tools/mod.hpp)
• tools::integral except for bool (tools/non_bool_integral.hpp)
• The number of nanoseconds that have elapsed since epoch (tools/now.hpp)
• Popcount (tools/popcount.hpp)
• $2^x$ (tools/pow2.hpp)
• $x^y \pmod{M}$ (tools/pow_mod.hpp)
• Pollard's rho algorithm (tools/prime_factorization.hpp)
• $x \cdot y \pmod{M}$ (tools/prod_mod.hpp)
• 128-bit unsigned integer (tools/uint128_t.hpp)

Code

// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/factorize

#include <iostream>
#include "tools/prime_factorization.hpp"

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  int Q;
  std::cin >> Q;
  for (int i = 0; i < Q; ++i) {
    long long a_i;
    std::cin >> a_i;

    const auto prime_factors = tools::prime_factorization(a_i);
    std::cout << prime_factors.size();
    for (const auto p : prime_factors) {
      std::cout << ' ' << p;
    }
    std::cout << '\n';
  }

  return 0;
}

#line 1 "tests/prime_factorization/regular.test.cpp"
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/factorize

#include <iostream>
#line 1 "tools/prime_factorization.hpp"



#include <algorithm>
#include <cassert>
#include <concepts>
#include <numeric>
#include <queue>
#include <tuple>
#include <utility>
#include <vector>
#line 1 "tools/floor_log2.hpp"



#line 1 "tools/bit_width.hpp"



#include <bit>
#line 6 "tools/bit_width.hpp"
#include <type_traits>
#line 1 "tools/is_signed.hpp"



#line 5 "tools/is_signed.hpp"

namespace tools {
  template <typename T>
  struct is_signed : std::is_signed<T> {};

  template <typename T>
  inline constexpr bool is_signed_v = tools::is_signed<T>::value;
}


#line 1 "tools/is_unsigned.hpp"



#line 5 "tools/is_unsigned.hpp"

namespace tools {
  template <typename T>
  struct is_unsigned : std::is_unsigned<T> {};

  template <typename T>
  inline constexpr bool is_unsigned_v = tools::is_unsigned<T>::value;
}


#line 1 "tools/make_unsigned.hpp"



#line 5 "tools/make_unsigned.hpp"

namespace tools {
  template <typename T>
  struct make_unsigned : std::make_unsigned<T> {};

  template <typename T>
  using make_unsigned_t = typename tools::make_unsigned<T>::type;
}


#line 1 "tools/non_bool_integral.hpp"



#line 1 "tools/integral.hpp"



#line 1 "tools/is_integral.hpp"



#line 5 "tools/is_integral.hpp"

namespace tools {
  template <typename T>
  struct is_integral : std::is_integral<T> {};

  template <typename T>
  inline constexpr bool is_integral_v = tools::is_integral<T>::value;
}


#line 5 "tools/integral.hpp"

namespace tools {
  template <typename T>
  concept integral = tools::is_integral_v<T>;
}


#line 7 "tools/non_bool_integral.hpp"

namespace tools {
  template <typename T>
  concept non_bool_integral = tools::integral<T> && !std::same_as<std::remove_cv_t<T>, bool>;
}


#line 12 "tools/bit_width.hpp"

namespace tools {
  namespace detail::bit_width {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::bit_width(x))) requires tools::is_unsigned_v<T> {
        return std::bit_width(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_width(T&& x) noexcept(noexcept(tools::detail::bit_width::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_width::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 6 "tools/floor_log2.hpp"

namespace tools {
  template <typename T>
  constexpr T floor_log2(T x) noexcept {
    assert(x > 0);
    return tools::bit_width(x) - 1;
  }
}


#line 1 "tools/is_prime.hpp"



#include <array>
#line 1 "tools/prod_mod.hpp"



#line 1 "tools/uint128_t.hpp"



#line 1 "tools/detail/int128_t_and_uint128_t.hpp"



#line 6 "tools/detail/int128_t_and_uint128_t.hpp"
#include <cstddef>
#include <cstdint>
#include <functional>
#line 10 "tools/detail/int128_t_and_uint128_t.hpp"
#include <limits>
#include <string>
#include <string_view>
#line 1 "tools/abs.hpp"



#include <cmath>
#line 7 "tools/abs.hpp"

namespace tools {
  namespace detail::abs {
    template <typename T>
    struct impl {
      constexpr decltype(auto) operator()(const T x) const noexcept(noexcept(std::abs(x))) {
        return std::abs(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) abs(T&& x) noexcept(noexcept(tools::detail::abs::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::abs::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/bit_ceil.hpp"



#line 12 "tools/bit_ceil.hpp"

namespace tools {
  namespace detail::bit_ceil {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr T operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr T operator()(const T x) const noexcept(noexcept(std::bit_ceil(x))) requires tools::is_unsigned_v<T> {
        return std::bit_ceil(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_ceil(T&& x) noexcept(noexcept(tools::detail::bit_ceil::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_ceil::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/bit_floor.hpp"



#line 12 "tools/bit_floor.hpp"

namespace tools {
  namespace detail::bit_floor {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr T operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr T operator()(const T x) const noexcept(noexcept(std::bit_floor(x))) requires tools::is_unsigned_v<T> {
        return std::bit_floor(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) bit_floor(T&& x) noexcept(noexcept(tools::detail::bit_floor::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::bit_floor::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/countr_zero.hpp"



#line 14 "tools/countr_zero.hpp"

namespace tools {
  namespace detail::countr_zero {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return std::min(impl<tools::make_unsigned_t<T>>{}(x), std::numeric_limits<T>::digits);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::countr_zero(x))) requires tools::is_unsigned_v<T> {
        return std::countr_zero(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) countr_zero(T&& x) noexcept(noexcept(tools::detail::countr_zero::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::countr_zero::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/gcd.hpp"



#line 7 "tools/gcd.hpp"

namespace tools {
  namespace detail::gcd {
    template <typename M, typename N>
    struct impl {
      constexpr decltype(auto) operator()(const M m, const N n) const noexcept(noexcept(std::gcd(m, n))) {
        return std::gcd(m, n);
      }
    };
  }

  template <typename M, typename N>
  constexpr decltype(auto) gcd(M&& m, N&& n) noexcept(noexcept(tools::detail::gcd::impl<std::remove_cvref_t<M>, std::remove_cvref_t<N>>{}(std::forward<M>(m), std::forward<N>(n)))) {
    return tools::detail::gcd::impl<std::remove_cvref_t<M>, std::remove_cvref_t<N>>{}(std::forward<M>(m), std::forward<N>(n));
  }
}


#line 1 "tools/has_single_bit.hpp"



#line 12 "tools/has_single_bit.hpp"

namespace tools {
  namespace detail::has_single_bit {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr bool operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr bool operator()(const T x) const noexcept(noexcept(std::has_single_bit(x))) requires tools::is_unsigned_v<T> {
        return std::has_single_bit(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) has_single_bit(T&& x) noexcept(noexcept(tools::detail::has_single_bit::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::has_single_bit::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 1 "tools/hash_combine.hpp"



#line 6 "tools/hash_combine.hpp"

// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.

// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

namespace tools {
  template <typename T>
  void hash_combine(std::size_t& seed, const T& v) {
    static const std::hash<T> hasher;
    static constexpr std::size_t k_mul = 0x9ddfea08eb382d69ULL;
    std::size_t a = (hasher(v) ^ seed) * k_mul;
    a ^= (a >> 47);
    std::size_t b = (seed ^ a) * k_mul;
    b ^= (b >> 47);
    seed = b * k_mul;
  }
}


#line 1 "tools/make_signed.hpp"



#line 5 "tools/make_signed.hpp"

namespace tools {
  template <typename T>
  struct make_signed : std::make_signed<T> {};

  template <typename T>
  using make_signed_t = typename tools::make_signed<T>::type;
}


#line 1 "tools/now.hpp"



#include <chrono>

namespace tools {
  inline long long now() {
    return std::chrono::duration_cast<std::chrono::nanoseconds>(std::chrono::high_resolution_clock::now().time_since_epoch()).count();
  }
}


#line 1 "tools/popcount.hpp"



#line 12 "tools/popcount.hpp"

namespace tools {
  namespace detail::popcount {
    template <tools::non_bool_integral T>
    struct impl {
      constexpr int operator()(const T x) const noexcept(noexcept(impl<tools::make_unsigned_t<T>>{}(x))) requires tools::is_signed_v<T> {
        assert(x >= 0);
        return impl<tools::make_unsigned_t<T>>{}(x);
      }
      constexpr int operator()(const T x) const noexcept(noexcept(std::popcount(x))) requires tools::is_unsigned_v<T> {
        return std::popcount(x);
      }
    };
  }

  template <typename T>
  constexpr decltype(auto) popcount(T&& x) noexcept(noexcept(tools::detail::popcount::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x)))) {
    return tools::detail::popcount::impl<std::remove_cvref_t<T>>{}(std::forward<T>(x));
  }
}


#line 31 "tools/detail/int128_t_and_uint128_t.hpp"

namespace tools {
  using uint128_t = unsigned __int128;
  using int128_t = __int128;

  template <>
  struct is_integral<tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<const tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<const tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<volatile tools::uint128_t> : std::true_type {};
  template <>
  struct is_integral<const volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_integral<const volatile tools::uint128_t> : std::true_type {};

  template <>
  struct is_signed<tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<const tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<const tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<volatile tools::uint128_t> : std::false_type {};
  template <>
  struct is_signed<const volatile tools::int128_t> : std::true_type {};
  template <>
  struct is_signed<const volatile tools::uint128_t> : std::false_type {};

  template <>
  struct is_unsigned<tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<const tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<const tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<volatile tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<volatile tools::uint128_t> : std::true_type {};
  template <>
  struct is_unsigned<const volatile tools::int128_t> : std::false_type {};
  template <>
  struct is_unsigned<const volatile tools::uint128_t> : std::true_type {};

  template <>
  struct make_signed<tools::int128_t> {
    using type = tools::int128_t;
  };
  template <>
  struct make_signed<tools::uint128_t> {
    using type = tools::int128_t;
  };
  template <>
  struct make_signed<const tools::int128_t> {
    using type = const tools::int128_t;
  };
  template <>
  struct make_signed<const tools::uint128_t> {
    using type = const tools::int128_t;
  };
  template <>
  struct make_signed<volatile tools::int128_t> {
    using type = volatile tools::int128_t;
  };
  template <>
  struct make_signed<volatile tools::uint128_t> {
    using type = volatile tools::int128_t;
  };
  template <>
  struct make_signed<const volatile tools::int128_t> {
    using type = const volatile tools::int128_t;
  };
  template <>
  struct make_signed<const volatile tools::uint128_t> {
    using type = const volatile tools::int128_t;
  };

  template <>
  struct make_unsigned<tools::int128_t> {
    using type = tools::uint128_t;
  };
  template <>
  struct make_unsigned<tools::uint128_t> {
    using type = tools::uint128_t;
  };
  template <>
  struct make_unsigned<const tools::int128_t> {
    using type = const tools::uint128_t;
  };
  template <>
  struct make_unsigned<const tools::uint128_t> {
    using type = const tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile tools::int128_t> {
    using type = volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<volatile tools::uint128_t> {
    using type = volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile tools::int128_t> {
    using type = const volatile tools::uint128_t;
  };
  template <>
  struct make_unsigned<const volatile tools::uint128_t> {
    using type = const volatile tools::uint128_t;
  };

  namespace detail::int128_t {
    constexpr tools::uint128_t parse_unsigned(const std::string_view s) noexcept {
      assert(!s.empty());
      tools::uint128_t x = 0;
      std::size_t i = s[0] == '+';
      if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
        for (i += 2; i < s.size(); ++i) {
          assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
          x <<= 4;
          if ('0' <= s[i] && s[i] <= '9') {
            x |= s[i] - '0';
          } else if ('a' <= s[i] && s[i] <= 'f') {
            x |= s[i] - 'a' + 10;
          } else {
            x |= s[i] - 'A' + 10;
          }
        }
      } else {
        for (; i < s.size(); ++i) {
          assert('0' <= s[i] && s[i] <= '9');
          x *= 10;
          x += s[i] - '0';
        }
      }
      return x;
    }

    constexpr tools::int128_t parse_signed(const std::string_view s) noexcept {
      assert(!s.empty());
      tools::int128_t x = 0;
      if (s[0] == '-') {
        std::size_t i = 1;
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x *= 16;
            if ('0' <= s[i] && s[i] <= '9') {
              x -= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x -= s[i] - 'a' + 10;
            } else {
              x -= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x -= s[i] - '0';
          }
        }
      } else {
        std::size_t i = s[0] == '+';
        if (i + 1 < s.size() && s[i] == '0' && (s[i + 1] == 'x' || s[i + 1] == 'X')) {
          for (i += 2; i < s.size(); ++i) {
            assert(('0' <= s[i] && s[i] <= '9') || ('a' <= s[i] && s[i] <= 'f') || ('A' <= s[i] && s[i] <= 'F'));
            x <<= 4;
            if ('0' <= s[i] && s[i] <= '9') {
              x |= s[i] - '0';
            } else if ('a' <= s[i] && s[i] <= 'f') {
              x |= s[i] - 'a' + 10;
            } else {
              x |= s[i] - 'A' + 10;
            }
          }
        } else {
          for (; i < s.size(); ++i) {
            assert('0' <= s[i] && s[i] <= '9');
            x *= 10;
            x += s[i] - '0';
          }
        }
      }
      return x;
    }
  }
}

#define UINT128_C(c) tools::detail::int128_t::parse_unsigned(#c)
#define INT128_C(c) tools::detail::int128_t::parse_signed(#c)

inline std::istream& operator>>(std::istream& is, tools::uint128_t& x) {
  std::string s;
  is >> s;
  x = tools::detail::int128_t::parse_unsigned(s);
  return is;
}
inline std::istream& operator>>(std::istream& is, tools::int128_t& x) {
  std::string s;
  is >> s;
  x = tools::detail::int128_t::parse_signed(s);
  return is;
}

inline std::ostream& operator<<(std::ostream& os, tools::uint128_t x) {
  std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else {
    s.push_back('0');
  }

  std::ranges::reverse(s);
  return os << s;
}
inline std::ostream& operator<<(std::ostream& os, tools::int128_t x) {
  std::string s;
  if (x > 0) {
    while (x > 0) {
      s.push_back('0' + x % 10);
      x /= 10;
    }
  } else if (x < 0) {
    while (x < 0) {
      s.push_back('0' + (-(x % 10)));
      x /= 10;
    }
    s.push_back('-');
  } else {
    s.push_back('0');
  }

  std::ranges::reverse(s);
  return os << s;
}

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
namespace std {
  template <>
  struct hash<tools::uint128_t> {
    std::size_t operator()(const tools::uint128_t& x) const {
      static const std::size_t seed = tools::now();

      std::size_t hash = seed;
      tools::hash_combine(hash, static_cast<std::uint64_t>(x >> 64));
      tools::hash_combine(hash, static_cast<std::uint64_t>(x & ((UINT128_C(1) << 64) - 1)));
      return hash;
    }
  };
  template <>
  struct hash<tools::int128_t> {
    std::size_t operator()(const tools::int128_t& x) const {
      static std::hash<tools::uint128_t> hasher;
      return hasher(static_cast<tools::uint128_t>(x));
    }
  };
}
#endif

namespace tools {
  template <>
  struct detail::abs::impl<tools::int128_t> {
    constexpr tools::int128_t operator()(const tools::int128_t& x) const noexcept {
      return x >= 0 ? x : -x;
    }
  };

#if defined(__GLIBCXX__) && defined(__STRICT_ANSI__)
  template <>
  struct detail::bit_ceil::impl<tools::uint128_t> {
    constexpr tools::uint128_t operator()(tools::uint128_t x) const noexcept {
      if (x <= 1) return 1;
      --x;
      x |= x >> 1;
      x |= x >> 2;
      x |= x >> 4;
      x |= x >> 8;
      x |= x >> 16;
      x |= x >> 32;
      x |= x >> 64;
      return ++x;
    }
  };

  template <>
  struct detail::bit_floor::impl<tools::uint128_t> {
    constexpr tools::uint128_t operator()(tools::uint128_t x) const noexcept {
      x |= x >> 1;
      x |= x >> 2;
      x |= x >> 4;
      x |= x >> 8;
      x |= x >> 16;
      x |= x >> 32;
      x |= x >> 64;
      return x & ~(x >> 1);
    }
  };

  template <>
  struct detail::bit_width::impl<tools::uint128_t> {
    constexpr int operator()(tools::uint128_t x) const noexcept {
      int w = 0;
      if (x & UINT128_C(0xffffffffffffffff0000000000000000)) {
        x >>= 64;
        w += 64;
      }
      if (x & UINT128_C(0xffffffff00000000)) {
        x >>= 32;
        w += 32;
      }
      if (x & UINT128_C(0xffff0000)) {
        x >>= 16;
        w += 16;
      }
      if (x & UINT128_C(0xff00)) {
        x >>= 8;
        w += 8;
      }
      if (x & UINT128_C(0xf0)) {
        x >>= 4;
        w += 4;
      }
      if (x & UINT128_C(0xc)) {
        x >>= 2;
        w += 2;
      }
      if (x & UINT128_C(0x2)) {
        x >>= 1;
        w += 1;
      }
      w += x;
      return w;
    }
  };

  template <>
  class detail::countr_zero::impl<tools::uint128_t> {
    using type = tools::uint128_t;
    static constexpr int shift = 120;
    static constexpr type magic = UINT128_C(0x01061438916347932a5cd9d3ead7b77f);
    static constexpr int ntz_table[255] = {
      128,   0,   1,  -1,   2,  -1,   8,  -1,   3,  -1,  15,  -1,   9,  -1,  22,  -1,
        4,  -1,  29,  -1,  16,  -1,  36,  -1,  10,  -1,  43,  -1,  23,  -1,  50,  -1,
        5,  -1,  33,  -1,  30,  -1,  57,  -1,  17,  -1,  64,  -1,  37,  -1,  71,  -1,
       11,  -1,  60,  -1,  44,  -1,  78,  -1,  24,  -1,  85,  -1,  51,  -1,  92,  -1,
       -1,   6,  -1,  20,  -1,  34,  -1,  48,  31,  -1,  -1,  69,  58,  -1,  -1,  90,
       18,  -1,  67,  -1,  65,  -1,  99,  -1,  38,  -1, 101,  -1,  72,  -1, 106,  -1,
       -1,  12,  -1,  40,  -1,  61,  -1,  82,  45,  -1,  -1, 103,  79,  -1, 113,  -1,
       -1,  25,  -1,  74,  86,  -1,  -1, 116,  -1,  52,  -1, 108,  -1,  93,  -1, 120,
      127,  -1,  -1,   7,  -1,  14,  -1,  21,  -1,  28,  -1,  35,  -1,  42,  -1,  49,
       -1,  32,  -1,  56,  -1,  63,  -1,  70,  -1,  59,  -1,  77,  -1,  84,  -1,  91,
       -1,  19,  -1,  47,  -1,  68,  -1,  89,  -1,  66,  -1,  98,  -1, 100,  -1, 105,
       -1,  39,  -1,  81,  -1, 102,  -1, 112,  -1,  73,  -1, 115,  -1, 107,  -1, 119,
      126,  -1,  13,  -1,  27,  -1,  41,  -1,  -1,  55,  62,  -1,  -1,  76,  83,  -1,
       -1,  46,  -1,  88,  -1,  97,  -1, 104,  -1,  80,  -1, 111,  -1, 114,  -1, 118,
      125,  -1,  26,  -1,  54,  -1,  75,  -1,  -1,  87,  96,  -1,  -1, 110,  -1, 117,
      124,  -1,  53,  -1,  -1,  95, 109,  -1, 123,  -1,  94,  -1, 122,  -1, 121
    };

  public:
    constexpr int operator()(const type& x) const noexcept {
      return ntz_table[static_cast<type>(magic * static_cast<type>(x & -x)) >> shift];
    }
  };

  namespace detail::gcd {
    template <>
    struct impl<tools::uint128_t, tools::uint128_t> {
      constexpr tools::uint128_t operator()(tools::uint128_t m, tools::uint128_t n) const noexcept {
        while (n != 0) {
          m %= n;
          std::swap(m, n);
        }
        return m;
      };
    };

    template <typename T>
    concept non_bool_integral_at_most_128bit = tools::non_bool_integral<T> && std::numeric_limits<T>::digits <= 128;
    template <typename T>
    concept non_bool_integral_at_most_64bit = tools::non_bool_integral<T> && std::numeric_limits<T>::digits <= 64;

    template <typename M, typename N> requires (
      (non_bool_integral_at_most_128bit<M> && non_bool_integral_at_most_128bit<N>)
      && !(non_bool_integral_at_most_64bit<M> && non_bool_integral_at_most_64bit<N>)
      && !(std::same_as<M, tools::uint128_t> && std::same_as<N, tools::uint128_t>)
    )
    struct impl<M, N> {
      constexpr std::common_type_t<M, N> operator()(const M m, const N n) const noexcept {
        return std::common_type_t<M, N>(
          tools::gcd(
            m >= 0 ? tools::uint128_t(m) : tools::uint128_t(-(m + 1)) + 1,
            n >= 0 ? tools::uint128_t(n) : tools::uint128_t(-(n + 1)) + 1
          )
        );
      }
    };
  }

  template <>
  struct detail::has_single_bit::impl<tools::uint128_t> {
    constexpr bool operator()(tools::uint128_t x) const noexcept {
      return x != 0 && (x & (x - 1)) == 0;
    }
  };

  template <>
  struct detail::popcount::impl<tools::uint128_t> {
    constexpr int operator()(tools::uint128_t x) const noexcept {
      x = (x & UINT128_C(0x55555555555555555555555555555555)) + (x >> 1 & UINT128_C(0x55555555555555555555555555555555));
      x = (x & UINT128_C(0x33333333333333333333333333333333)) + (x >> 2 & UINT128_C(0x33333333333333333333333333333333));
      x = (x & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f)) + (x >> 4 & UINT128_C(0x0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f0f));
      x = (x & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff)) + (x >> 8 & UINT128_C(0x00ff00ff00ff00ff00ff00ff00ff00ff));
      x = (x & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff)) + (x >> 16 & UINT128_C(0x0000ffff0000ffff0000ffff0000ffff));
      x = (x & UINT128_C(0x00000000ffffffff00000000ffffffff)) + (x >> 32 & UINT128_C(0x00000000ffffffff00000000ffffffff));
      x = (x & UINT128_C(0x0000000000000000ffffffffffffffff)) + (x >> 64 & UINT128_C(0x0000000000000000ffffffffffffffff));
      return x;
    }
  };
#endif
}


#line 5 "tools/uint128_t.hpp"


#line 5 "tools/prod_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 prod_mod(const T1 x, const T2 y, const T3 m) {
    using u128 = tools::uint128_t;
    u128 prod_mod = u128(x >= 0 ? x : -x) * u128(y >= 0 ? y : -y) % u128(m);
    if ((x >= 0) ^ (y >= 0)) prod_mod = u128(m) - prod_mod;
    return prod_mod;
  }
}


#line 1 "tools/pow_mod.hpp"



#line 1 "tools/mod.hpp"



#line 7 "tools/mod.hpp"

namespace tools {
  template <tools::non_bool_integral M, tools::non_bool_integral N>
  constexpr std::common_type_t<M, N> mod(const M a, const N b) noexcept {
    assert(b != 0);

    using UM = tools::make_unsigned_t<M>;
    using UN = tools::make_unsigned_t<N>;
    const UM ua = a >= 0 ? a : static_cast<UM>(-(a + 1)) + 1;
    const UN ub = b >= 0 ? b : static_cast<UN>(-(b + 1)) + 1;
    auto r = ua % ub;
    if (a < 0 && r > 0) {
      r = ub - r;
    }
    return r;
  }
}


#line 6 "tools/pow_mod.hpp"

namespace tools {

  template <typename T1, typename T2, typename T3>
  constexpr T3 pow_mod(const T1 x, T2 n, const T3 m) {
    if (m == 1) return 0;
    T3 r = 1;
    T3 y = tools::mod(x, m);
    while (n > 0) {
      if ((n & 1) > 0) {
        r = tools::prod_mod(r, y, m);
      }
      y = tools::prod_mod(y, y, m);
      n /= 2;
    }
    return r;
  }
}


#line 7 "tools/is_prime.hpp"

namespace tools {

  constexpr bool is_prime(const unsigned long long n) {
    constexpr std::array<unsigned long long, 7> bases = {2, 325, 9375, 28178, 450775, 9780504, 1795265022};

    if (n <= 1) return false;
    if (n == 2) return true;
    if (n % 2 == 0) return false;

    auto d = n - 1;
    for (; d % 2 == 0; d /= 2);

    for (const auto a : bases) {
      if (a % n == 0) return true;

      auto power = d;
      auto target = tools::pow_mod(a, power, n);

      bool is_composite = true;
      if (target == 1) is_composite = false;
      for (; is_composite && power != n - 1; power *= 2, target = tools::prod_mod(target, target, n)) {
        if (target == n - 1) is_composite = false;
      }

      if (is_composite) {
        return false;
      }
    }

    return true;
  }
}


#line 1 "tools/pow2.hpp"



#line 7 "tools/pow2.hpp"

namespace tools {
  template <tools::integral T>
  constexpr T pow2(const T x) noexcept {
    assert(0 <= x && x < std::numeric_limits<T>::digits);
    return T(1) << x;
  }
}


#line 16 "tools/prime_factorization.hpp"

namespace tools {
  template <std::integral T>
  std::vector<T> prime_factorization(T n) {
    assert(n >= 1);
    std::vector<T> res;

    if (n == 1) return res;

    std::queue<std::pair<T, T>> factors;
    factors.emplace(n, 1);
    while (!factors.empty()) {
      const auto [factor, occurrences] = factors.front();
      factors.pop();
      if (tools::is_prime(factor)) {
        for (T i = 0; i < occurrences; ++i) {
          res.push_back(factor);
        }
      } else {
        const T m = tools::pow2((tools::floor_log2(factor) + 1) / 8);
        for (T c = 1; ; ++c) {
          const auto f = [&](T& x) {
            x = tools::prod_mod(x, x, factor);
            // (x + c) mod factor s.t. 0 <= x < factor and 1 <= c < factor
            if (x >= factor - c) {
              x -= factor - c;
            } else {
              x += c;
            }
          };
          T y = 2;
          T r = 1;
          T q = 1;
          T x, g, ys;
          do {
            x = y;
            for (T i = 0; i < r; ++i) {
              f(y);
            }
            T k = 0;
            do {
              ys = y;
              for (T i = 0; i < std::min<T>(m, r - k); ++i) {
                f(y);
                q = tools::prod_mod(q, std::max(x, y) - std::min(x, y), factor);
              }
              g = std::gcd(q, factor);
              k += m;
            } while (k < r && g == 1);
            r *= 2;
          } while (g == 1);
          if (g == factor) {
            do {
              f(ys);
              g = std::gcd(std::max(x, ys) - std::min(x, ys), factor);
            } while (g == 1);
          }
          if (g < factor) {
            T h = factor / g;
            std::tie(g, h) = std::minmax({g, h});
            T n = 1;
            while (h % g == 0) {
              h /= g;
              ++n;
            }
            factors.emplace(g, occurrences * n);
            if (h > 1) factors.emplace(h, occurrences);
            break;
          }
        }
      }
    }

    std::ranges::sort(res);
    return res;
  }
}


#line 5 "tests/prime_factorization/regular.test.cpp"

int main() {
  std::cin.tie(nullptr);
  std::ios_base::sync_with_stdio(false);

  int Q;
  std::cin >> Q;
  for (int i = 0; i < Q; ++i) {
    long long a_i;
    std::cin >> a_i;

    const auto prime_factors = tools::prime_factorization(a_i);
    std::cout << prime_factors.size();
    for (const auto p : prime_factors) {
      std::cout << ' ' << p;
    }
    std::cout << '\n';
  }

  return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++	4295098369_00	AC	6 ms	4 MB
g++	999381247093216751_00	AC	184 ms	3 MB
g++	big2_00	AC	47 ms	4 MB
g++	big2_01	AC	49 ms	4 MB
g++	big2_02	AC	49 ms	4 MB
g++	big2_worse_00	AC	98 ms	3 MB
g++	big_semiprime_gen_00	AC	47 ms	4 MB
g++	big_semiprime_gen_01	AC	46 ms	4 MB
g++	big_semiprime_random_00	AC	38 ms	4 MB
g++	big_semiprime_random_01	AC	40 ms	4 MB
g++	carmichael_00	AC	8 ms	4 MB
g++	example_00	AC	4 ms	4 MB
g++	fixed_RNG_buster_00	AC	4 ms	4 MB
g++	hack00_00	AC	4 ms	4 MB
g++	max_00	AC	7 ms	4 MB
g++	pow2_00	AC	7 ms	3 MB
g++	pow2_01	AC	8 ms	4 MB
g++	pow2_02	AC	10 ms	4 MB
g++	prime_test_special_00	AC	4 ms	4 MB
g++	prime_test_special_01	AC	5 ms	4 MB
g++	prime_test_special_02	AC	6 ms	4 MB
g++	prime_test_special_03	AC	5 ms	3 MB
g++	prime_test_special_bug_00	AC	4 ms	4 MB
g++	prime_test_special_bug_01	AC	4 ms	4 MB
g++	prime_test_special_bug_02	AC	4 ms	4 MB
g++	random_00	AC	5 ms	4 MB
g++	random_01	AC	5 ms	3 MB
g++	random_02	AC	5 ms	4 MB
g++	small_00	AC	4 ms	4 MB
g++	small_01	AC	4 ms	3 MB
g++	small_02	AC	4 ms	4 MB
clang++	4295098369_00	AC	6 ms	4 MB
clang++	999381247093216751_00	AC	214 ms	4 MB
clang++	big2_00	AC	54 ms	4 MB
clang++	big2_01	AC	56 ms	4 MB
clang++	big2_02	AC	56 ms	4 MB
clang++	big2_worse_00	AC	113 ms	4 MB
clang++	big_semiprime_gen_00	AC	53 ms	4 MB
clang++	big_semiprime_gen_01	AC	53 ms	4 MB
clang++	big_semiprime_random_00	AC	44 ms	4 MB
clang++	big_semiprime_random_01	AC	46 ms	4 MB
clang++	carmichael_00	AC	8 ms	4 MB
clang++	example_00	AC	4 ms	4 MB
clang++	fixed_RNG_buster_00	AC	4 ms	4 MB
clang++	hack00_00	AC	4 ms	4 MB
clang++	max_00	AC	7 ms	4 MB
clang++	pow2_00	AC	8 ms	4 MB
clang++	pow2_01	AC	8 ms	4 MB
clang++	pow2_02	AC	11 ms	4 MB
clang++	prime_test_special_00	AC	5 ms	4 MB
clang++	prime_test_special_01	AC	5 ms	4 MB
clang++	prime_test_special_02	AC	6 ms	4 MB
clang++	prime_test_special_03	AC	4 ms	4 MB
clang++	prime_test_special_bug_00	AC	4 ms	4 MB
clang++	prime_test_special_bug_01	AC	4 ms	4 MB
clang++	prime_test_special_bug_02	AC	4 ms	4 MB
clang++	random_00	AC	5 ms	4 MB
clang++	random_01	AC	5 ms	4 MB
clang++	random_02	AC	5 ms	4 MB
clang++	small_00	AC	4 ms	4 MB
clang++	small_01	AC	4 ms	4 MB
clang++	small_02	AC	4 ms	4 MB

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