Problem:
Score : 400 points

Problem Statement

You are given positive integers N and M.

How many sequences a of length N consisting of positive integers satisfy a_1 \times a_2 \times ... \times a_N = M? Find the count modulo 10^9+7.

Here, two sequences a' and a'' are considered different when there exists some i such that a_i' \neq a_i''.

Constraints

All values in input are integers.

1 \leq N \leq 10^5

1 \leq M \leq 10^9

Input

Input is given from Standard Input in the following format:

N M

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo 10^9 + 7.

Sample Input 1

2 6

Sample Output 1

4

Four sequences satisfy the condition: \{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\} and \{6, 1\}.

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(clippy::many_single_char_names)]

// language: Rust(1.42.0)
// check available crates on AtCoder at "https://atcoder.jp/contests/language-test-202001"
// My Library Repo' is at "https://github.com/Loptall/prcn_lib"

/*
    そう甘くはないんだな〜w
*/

// from std...
use std::cmp::{
    max, min,
    Ordering,
    Ordering::{Equal, Greater, Less},
    Reverse,
};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::convert::TryInto;
use std::mem::swap;
use std::num::{ParseIntError, NonZeroU32};
use std::ops::*;
use std::process::exit;
use std::{f32, f64, i128, i16, i32, i64, i8, isize, u128, u16, u32, u64, u8, usize};

// Dep' crates are...
use itertools::*;
use itertools_num::*;
use lazy_static::lazy_static;
use maplit::{btreemap, btreeset, hashmap, hashset};
use num_bigint::{BigInt, BigUint};
use num_complex::Complex;
use num_integer::{binomial, multinomial, gcd, lcm, Integer};
use num_rational::Rational;
use num_traits::{clamp, one, pow, zero, Num, NumAssignOps, NumOps, One, Pow, Zero};
use permutohedron::Heap;
use petgraph::unionfind::UnionFind;
use proconio::{fastout, input, is_stdin_empty, marker::*};
use rand::random;

pub const MOD: usize = 1000000007;
pub const MAX: usize = 1usize << 63usize;
pub const MIN: i64 = i64::MIN;
pub const PI: f64 = f64::consts::PI;
pub const A: u8 = 'A' as u8;

/// 2以上で`x`の約数であるものの内最小であるものを求める
fn firstfac(x: usize) -> usize {
    if x % 2 == 0 {
        return 2;
    };
    for n in (3..).step_by(2).take_while(|m| m * m <= x) {
        if x % n == 0 {
            return n;
        };
    }
    x
}
/// 試し割りによる素数判定です
///
/// `O(√n)`
pub fn is_prime(n: usize) -> bool {
    if n <= 1 {
        return false;
    }
    firstfac(n) == n
}
fn exp(n: &mut usize, d: usize) -> (usize, usize) {
    let mut c = 0usize;
    (
        d,
        loop {
            if *n % d != 0 {
                break c;
            }
            *n /= d;
            c += 1;
        },
    )
}
pub fn factorization(mut n: usize) -> Vec<(usize, usize)> {
    let mut primes = Vec::new();
    let (d, e) = exp(&mut n, 2);
    if e > 0 {
        primes.push((d, e));
    }
    let m = n;
    for i in (3..).take_while(|x| *x * *x <= m).step_by(2) {
        let (d, e) = exp(&mut n, i);
        if e > 0 {
            primes.push((d, e));
        }
    }
    if n > 1 {
        match primes.iter().enumerate().find(|x| (x.1).0 == n) {
            Some((i, _)) => primes[i].1 += 1,
            None => primes.push((n, 1)),
        }
    }
    primes
}

/// n % m
/// ただし答えが負になる場合は余分にmを足すことで一意な値を保証
///
/// # Panic
/// 異なるmod間での演算をattemptした時
fn compensated_rem(n: i64, m: usize) -> i64 {
    match n % m as i64 {
        r if r >= 0 => r,
        r => r + m as i64,
    }
}
#[derive(Debug, Copy, Clone, Eq, PartialEq)]
pub enum Modulo {
    Static(NonZeroU32),
    Dynamic,
}
impl Modulo {
    pub fn get(&self) -> Option<u32> {
        match self {
            Modulo::Static(nz) => Some(nz.get()),
            Modulo::Dynamic => None,
        }
    }
}
/// `ModInt -> PrimiteveInt` への暗黙のキャストは行わない!
/// (get関数を提供するのでそれ使ってどうぞ)
///
/// `PrimitiveInt -> ModInt` は許可する
#[derive(Debug, Clone, Copy)]
pub struct ModInt {
    num: i64,
    _modulo: Modulo,
}
impl Into<usize> for ModInt {
    fn into(self) -> usize {
        self.get() as usize
    }
}
pub trait IntoModInt: Copy {
    fn to_mint<M: TryInto<u32> + Copy>(self, modulo: M) -> ModInt;
}
macro_rules ! impl_into_mint {($ ($ t : ty ) ,* ) => {$ (impl IntoModInt for $ t {fn to_mint < M : TryInto < u32 > + Copy > (self , modulo : M ) -> ModInt {ModInt :: new (self , modulo ) } } ) * } ; }
impl_into_mint!(usize, u8, u16, u32, u64, isize, i8, i16, i32, i64);
impl PartialEq for ModInt {
    fn eq(&self, other: &Self) -> bool {
        if !check_mod_eq(self, other).1 {
            panic!("cannot compare these values because they have different modulo number")
        }
        self.get() == other.num
    }
}
impl PartialOrd for ModInt {
    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
        if !check_mod_eq(self, other).1 {
            None
        } else {
            Some(self.get().cmp(&other.num))
        }
    }
}
fn check_mod_eq(a: &ModInt, b: &ModInt) -> (NonZeroU32, bool) {
    match (a._modulo, b._modulo) {
        (Modulo::Static(a), Modulo::Static(b)) => {
            if a == b {
                (a, true)
            } else {
                (unsafe { NonZeroU32::new_unchecked(1) }, false)
            }
        }
        (Modulo::Static(m), Modulo::Dynamic) | (Modulo::Dynamic, Modulo::Static(m)) => (m, true),
        (Modulo::Dynamic, Modulo::Dynamic) => (unsafe { NonZeroU32::new_unchecked(1) }, false),
    }
}
impl ModInt {
    /// always `_modulo > num >= 0 && _modulo >= 1`
    pub fn new<N: TryInto<i64>, M: TryInto<u32> + Copy>(n: N, m: M) -> Self {
        let m = NonZeroU32::new(m.try_into().ok().expect("modulo number may be wrong")).unwrap();
        let r = n
            .try_into()
            .ok()
            .expect("modulo number maybe over i64 range");
        let num = compensated_rem(r, m.get() as usize);
        Self {
            num,
            _modulo: Modulo::Static(m),
        }
    }
    /// get inner value
    pub fn get(&self) -> i64 {
        self.num
    }
    /// mod of modint
    ///
    /// # Panic
    /// if variant is Modulo::Dynamic
    pub fn get_mod(&self) -> usize {
        self._modulo.get().unwrap() as usize
    }
    /// return the power of self with mod, using binary powering method
    /// cannot use of Dynamic type mod Self
    pub fn pow_mod(&self, mut exp: usize) -> Self {
        let mut res = 1;
        let mut base = self.get() as usize;
        let m = self.get_mod();
        while exp > 0 {
            if exp & 1 != 0 {
                res *= base;
                res %= m;
            }
            base *= base;
            base %= m;
            exp >>= 1;
        }
        Self::new(res, self.get_mod())
    }
    /// `a / b == a * b^(-1)` となる `b^(-1)` を求める
    pub fn inv(&self) -> i64 {
        let x = self.get().extended_gcd(&(self.get_mod() as i64)).x;
        compensated_rem(x, self.get_mod())
    }
}
impl Add<Self> for ModInt {
    type Output = Self;
    fn add(self, rhs: Self) -> Self::Output {
        let c = check_mod_eq(&self, &rhs);
        if !c.1 {
            panic!("modulo between two instance is different!")
        }
        let r = self.get() + rhs.num;
        Self {
            num: if r >= self.get_mod() as i64 {
                r - c.0.get() as i64
            } else {
                r
            },
            _modulo: Modulo::Static(c.0),
        }
    }
}
impl AddAssign<Self> for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}
impl Sub<Self> for ModInt {
    type Output = Self;
    fn sub(self, rhs: Self) -> Self::Output {
        let c = check_mod_eq(&self, &rhs);
        if !c.1 {
            panic!("modulo between two instance is different!")
        }
        let num = compensated_rem(self.get() - rhs.get(), c.0.get() as usize);
        Self {
            num,
            _modulo: Modulo::Static(c.0),
        }
    }
}
impl SubAssign<Self> for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}
impl Mul<Self> for ModInt {
    type Output = Self;
    fn mul(self, rhs: Self) -> Self::Output {
        let c = check_mod_eq(&self, &rhs);
        if !c.1 {
            panic!("modulo between two instance is different!")
        }
        let num = compensated_rem(self.get() * rhs.get(), c.0.get() as usize);
        Self {
            num,
            _modulo: Modulo::Static(c.0),
        }
    }
}
impl MulAssign<Self> for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs
    }
}
impl Div<Self> for ModInt {
    type Output = Self;
    fn div(self, rhs: Self) -> Self::Output {
        let c = check_mod_eq(&self, &rhs);
        if !c.1 {
            panic!("modulo between two instance is different!")
        }
        Self {
            num: self.get() * rhs.inv() % c.0.get() as i64,
            _modulo: Modulo::Static(c.0),
        }
    }
}
impl DivAssign<Self> for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}
impl Rem for ModInt {
    type Output = Self;
    fn rem(self, rhs: Self) -> Self::Output {
        let c = check_mod_eq(&self, &rhs);
        if !c.1 {
            panic!("modulo between two instance is different!")
        }
        Self {
            num: self.num % rhs.num,
            _modulo: Modulo::Static(c.0),
        }
    }
}
impl RemAssign for ModInt {
    fn rem_assign(&mut self, rhs: Self) {
        *self = *self % rhs
    }
}
impl Zero for ModInt {
    fn zero() -> Self {
        ModInt {
            num: 0,
            _modulo: Modulo::Dynamic,
        }
    }
    fn is_zero(&self) -> bool {
        self.num == 0
    }
}
impl One for ModInt {
    fn one() -> Self {
        ModInt {
            num: 1,
            _modulo: Modulo::Dynamic,
        }
    }
    fn is_one(&self) -> bool {
        self.num == 1
    }
}
impl Num for ModInt {
    type FromStrRadixErr = ParseIntError;
    fn from_str_radix(str: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr> {
        let num = str
            .chars()
            .rev()
            .enumerate()
            .map(|(i, b)| radix.pow(i as u32) as i64 * b.to_digit(radix).unwrap() as i64)
            .sum::<i64>();
        Ok(ModInt {
            num,
            _modulo: Modulo::Dynamic,
        })
    }
}
impl Pow<usize> for ModInt {
    type Output = Self;
    fn pow(mut self, mut exp: usize) -> Self::Output {
        if exp == 0 {
            return Self::one();
        }
        while exp & 1 == 0 {
            self = self * self;
            exp >>= 1;
        }
        if exp == 1 {
            return self;
        }
        let mut acc = self;
        while exp > 1 {
            exp >>= 1;
            self = self * self;
            if exp & 1 == 1 {
                acc = acc * self;
            }
        }
        acc
    }
}
macro_rules ! impl_ops_between_mint_and_primitive {($ ($ t : ty ) ,* ) => {$ (impl Add <$ t > for ModInt {type Output = Self ; fn add (self , rhs : $ t ) -> Self :: Output {self + Self :: new (rhs as i64 , self . get_mod () ) } } impl AddAssign <$ t > for ModInt {fn add_assign (& mut self , rhs : $ t ) {* self = * self + rhs ; } } impl Sub <$ t > for ModInt {type Output = Self ; fn sub (self , rhs : $ t ) -> Self :: Output {self - Self :: new (rhs as i64 , self . get_mod () ) } } impl SubAssign <$ t > for ModInt {fn sub_assign (& mut self , rhs : $ t ) {* self = * self - rhs ; } } impl Mul <$ t > for ModInt {type Output = Self ; fn mul (self , rhs : $ t ) -> Self :: Output {self * Self :: new (rhs as i64 , self . get_mod () ) } } impl MulAssign <$ t > for ModInt {fn mul_assign (& mut self , rhs : $ t ) {* self = * self * rhs ; } } impl Div <$ t > for ModInt {type Output = Self ; fn div (self , rhs : $ t ) -> Self :: Output {self / Self :: new (rhs as i64 , self . get_mod () ) } } impl DivAssign <$ t > for ModInt {fn div_assign (& mut self , rhs : $ t ) {* self = * self / rhs ; } } ) * } ; }
impl_ops_between_mint_and_primitive!(usize, u8, u16, u32, u64, isize, i8, i16, i32, i64);

pub trait PartialBinomialCoefficient {
    fn partial_binomial(&self, n: usize, k: usize) -> Option<ModInt>;
}
pub trait BinomialCoefficient: PartialBinomialCoefficient {
    /// `n C k`
    fn binomial(&self, n: usize, k: usize) -> ModInt {
        self.partial_binomial(n, k).unwrap()
    }
}
/// Binomial Coefficient Table with DP
/// 二項係数を`O(1)`で計算するためのテーブル
///
/// factrial = [1, 1, 2, 6, 24, 120, ...],
///
/// `1 <= k <= n <= 10^7` 程度
pub struct BCTDP {
    _modulo: NonZeroU32,
    factorial: Vec<ModInt>,
    inverse: Vec<ModInt>,
    factorial_inverse: Vec<ModInt>,
}
impl BCTDP {
    /// 初期化
    ///
    /// DPを用いて `O(n log m)`
    /// 割り算を用いるので `log m` がつく
    ///
    /// `1 <= k <= n <= 10^7` 程度
    pub fn new(n: usize, modulo: usize) -> BCTDP {
        let mut factorial = vec![ModInt::new(1, modulo), ModInt::new(1, modulo)];
        factorial.reserve_exact(n);
        let mut inverse = vec![ModInt::new(0, modulo), ModInt::new(1, modulo)];
        inverse.reserve_exact(n);
        let mut factorial_inverse = vec![ModInt::new(1, modulo), ModInt::new(1, modulo)];
        factorial_inverse.reserve_exact(n);
        for i in 2..=n {
            factorial.push(factorial[i - 1] * i);
            inverse.push(modulo.to_mint(modulo) - inverse[modulo % i] * (modulo / i));
            factorial_inverse.push(factorial_inverse[i - 1] * inverse[i]);
        }
        Self {
            _modulo: NonZeroU32::new(modulo as u32).unwrap(),
            factorial,
            inverse,
            factorial_inverse,
        }
    }
    pub fn get_mod(&self) -> usize {
        self._modulo.get() as usize
    }
    pub fn factorial(&self, n: usize) -> ModInt {
        self.factorial[n]
    }
    pub fn factorial_inverse(&self, n: usize) -> ModInt {
        self.factorial_inverse[n]
    }
    /// `n` の mod self._modulo における逆元
    pub fn inv(&self, n: usize) -> ModInt {
        self.inverse[n]
    }
}
impl PartialBinomialCoefficient for BCTDP {
    fn partial_binomial(&self, n: usize, k: usize) -> Option<ModInt> {
        Some(if n < k {
            ModInt::zero()
        } else {
            self.factorial[n] * self.factorial_inverse[k] * self.factorial_inverse[n - k]
        })
    }
}
impl BinomialCoefficient for BCTDP {}
/// `n` が固定値のときに有効
/// `(n(固定値), mod, _[i] = n C i)`
///
/// 初期化: `O(n)`
///
/// `1 <= n <= 10^9 && 1 <= k <= 10^7` 程度
pub struct BCTholdN(usize, NonZeroU32, Vec<ModInt>);
impl BCTholdN {
    pub fn new(mut n: usize, m: usize) -> Self {
        let size = n;
        let mut c = vec![ModInt::new(1, m), ModInt::new(n, m)];
        c.reserve_exact(n + 1);
        for i in 2..=n {
            n -= 1;
            let prev = *c.last().unwrap();
            c.push(prev * n / i);
        }
        Self(size, NonZeroU32::new(m as u32).unwrap(), c)
    }
}
impl PartialBinomialCoefficient for BCTholdN {
    /// #Panic
    ///
    /// self.0 == _n でないとき
    fn partial_binomial(&self, _n: usize, k: usize) -> Option<ModInt> {
        if _n != self.0 {
            None
        } else {
            Some(self.2[k])
        }
    }
}
/// `n, k` の2変数についての `n C k` の表を作る
///
/// `1 <= k <= n <= 2000` 程度
pub struct BCTSmallNK {
    n: usize,
    _modulo: NonZeroU32,
    dp: Vec<Vec<ModInt>>,
}
impl BCTSmallNK {
    pub fn new(n: usize, modulo: usize) -> Self {
        let mut dp = vec![vec![ModInt::new(0, modulo); n + 1]; n + 1];
        dp[0][0] = 1.to_mint(modulo);
        for i in 1..n {
            dp[i][0] = 1.to_mint(modulo);
            for j in 1..n {
                dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j];
            }
        }
        Self {
            n,
            _modulo: NonZeroU32::new(modulo as u32).unwrap(),
            dp,
        }
    }
    pub fn size(&self) -> usize {
        self.n
    }
    pub fn get_mod(&self) -> usize {
        self._modulo.get() as usize
    }
}
impl PartialBinomialCoefficient for BCTSmallNK {
    fn partial_binomial(&self, n: usize, k: usize) -> Option<ModInt> {
        if n > self.size() || k > self.size() {
            panic!("n or k is too large, compere to dp table!")
        }
        Some(self.dp[n][k])
    }
}
impl BinomialCoefficient for BCTSmallNK {}

pub fn combination(n: ModInt, k: usize) -> ModInt {
    if k > n.get() as usize {
        panic!("n < k, where n in ModInt, k in usize, so cannot calculate n C k")
    }
    permutation(n, k) / k.factorial()
}
pub fn combination_with_table<T: BinomialCoefficient>(table: &T, n: usize, k: usize) -> ModInt {
    table.binomial(n, k)
}

/// `n P k` を `O(k)` で
///
/// 内部はfallingをラップしているだけ
pub fn permutation<T: Factoriable>(n: T, k: usize) -> T {
    n.falling(k)
}
pub fn permutation_with_table(table: &BCTDP, n: usize, k: usize) -> ModInt {
    if k > n {
        ModInt::new(0, table.get_mod())
    } else {
        table.factorial(n) * table.factorial_inverse(n - k)
    }
}

pub trait Factoriable: Sized + NumOps + NumAssignOps + Copy + TryInto<usize> {
    fn falling(self, take: usize) -> Self;
    fn rising(self, take: usize) -> Self;
    fn factorial(self) -> Self {
        self.falling(self.try_into().ok().unwrap())
    }
}
macro_rules ! impl_factorialbe {($ ($ t : ty ) ,* ) => {$ (impl Factoriable for $ t {fn falling (self , take : usize ) -> Self {let mut res = Self :: one () ; let mut c = self ; for _ in 0 .. take {res *= c ; c -= Self :: one () ; } res } fn rising (self , take : usize ) -> Self {let mut res = Self :: one () ; let mut c = self ; for _ in 0 .. take {res *= c ; c += 1 ; } res } } ) * } ; }
impl_factorialbe!(usize, u8, u16, u32, u64, isize, i8, i16, i32, i64, ModInt);

// code...
// #[fastout]
fn main() {
    input! {
        n: usize, m: usize
    }

    let f = factorization(m);
    // dbg!(&f);
    let mut ans = ModInt::new(1, MOD);

    let tbl = BCTDP::new(1000000, MOD);

    for e in f.iter().map(|x| x.1) {
        let p = combination_with_table(&tbl, n + e - 1, e);
        // dbg!(p);
        ans *= p;
    }

    println!("{}", ans.get());
}