The aim of the kata is to decompose `n!` (factorial n) into its prime factors.

Examples:
```
n = 12; decomp(12) -> "2^10 * 3^5 * 5^2 * 7 * 11"
since 12! is divisible by 2 ten times, by 3 five times, by 5 two times and by 7 and 11 only once.

n = 22; decomp(22) -> "2^19 * 3^9 * 5^4 * 7^3 * 11^2 * 13 * 17 * 19"

n = 25; decomp(25) -> 2^22 * 3^10 * 5^6 * 7^3 * 11^2 * 13 * 17 * 19 * 23
```

Prime numbers should be in increasing order. When the exponent of a prime is 1 don't put the exponent.

Notes

- the function is `decomp(n)` and should return the decomposition of `n!` into its prime factors in increasing order of the primes, as a string.
- factorial can be a very big number (`4000! has 12674 digits`, n will go from 300 to 4000).
- In Fortran - as in any other language - the returned string is not permitted to contain any redundant trailing whitespace: you can use `dynamically allocated character strings`.

Source: codewars | Difficulty: EASY
Tags: Fundamentals

Starter code: def decomp(n):
	

Solutions:
from collections import defaultdict

def dec(n):
	decomp = defaultdict(lambda : 0)
	i = 2
	while n > 1:
		while n % i == 0:
			n /= i
			decomp[i] += 1
		i += 1
	return decomp

def decomp(n):
	ans = defaultdict(lambda : 0)
	for i in range(2, n + 1):
		for (key, value) in dec(i).items():
			ans[key] += value
	return ' * '.join(('{}^{}'.format(x, y) if y > 1 else str(x) for (x, y) in sorted(ans.items())))


def decomp(chisl):
	div = [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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001]
	exp = [0] * len(div)
	otvet = []
	chislo = 1
	for i in range(1, chisl + 1):
		for ii in range(len(div)):
			while i % div[ii] == 0:
				i /= div[ii]
				exp[ii] += 1
	for ii in range(len(div)):
		if exp[ii] > 1:
			otvet += [str(div[ii]) + '^' + str(exp[ii])]
		if exp[ii] == 1:
			otvet += [str(div[ii])]
	return ' * '.join(otvet)


def decomp(n):
	out = []
	for d in range(2, n + 1):
		if d == 2 or (d & 1 and d > 2 and all((d % i for i in range(3, 1 + int(d ** 0.5), 2)))):
			(a, x) = (0, n)
			while x >= d:
				x //= d
				a += x
			out.append((d, a))
	return ' * '.join([f'{i}^{j}' if j > 1 else str(i) for (i, j) in out])


def decomp(n):
	f = {}
	for i in range(2, n + 1):
		for j in range(2, int(i ** 0.5) + 1):
			while i % j == 0:
				i = i // j
				if j in f:
					f[j] += 1
				else:
					f[j] = 1
		if i != 1:
			if i in f:
				f[i] += 1
			else:
				f[i] = 1
	return ' * '.join(['{}^{}'.format(i, f[i]) if f[i] > 1 else str(i) for i in sorted(f)])


from itertools import count
dec = [[], [], [1]]
primes = [2]

def genPrimes():
	for x in count(3, 2):
		sq = int(x ** 0.5)
		if all((p > sq or x % p for p in primes)):
			yield x
primGen = genPrimes()

def genPrimesUpTo(n):
	while primes[-1] < n:
		primes.append(next(primGen))

def genPrimeDivs(n):
	genPrimesUpTo(n)
	dec.append(dec[-1][:])
	for (i, p) in enumerate(primes):
		while not n % p:
			n //= p
			while i >= len(dec[-1]):
				dec[-1].append(0)
			dec[-1][i] += 1
			if n < 2:
				break

def decomp(n):
	while len(dec) <= n:
		genPrimeDivs(len(dec))
	return ' * '.join((str(p) if c == 1 else '{}^{}'.format(p, c) for (p, c) in zip(primes, dec[n])))


def decomp(n):
	result = []
	for i in primes():
		if i > n:
			break
		count = 0
		for j in range(2, n + 1):
			while j % i == 0:
				count += 1
				j = j / i
		if count > 1:
			result.append(str(i) + '^' + str(count))
		else:
			result.append(str(i))
	return ' * '.join(result)

def primes():
	yield 2
	it = odd()
	while True:
		n = next(it)
		yield n
		it = filter(lambda x, n=n: x % n > 0, it)

def odd():
	n = 1
	while True:
		n += 2
		yield n


import math
import numpy as np

def decomp(n):

	def rt(x):
		if x % 2 == 0:
			return [2, int(x / 2)]
		for i in range(3, math.floor(math.sqrt(x)) + 1, 2):
			if x % i == 0:
				return [i, int(x / i)]
		return [x, 1]
	leftovers = np.arange(2, n + 1, 1)
	facs = [0] * (n - 1)
	while leftovers != []:
		newlefts = []
		for l in leftovers:
			[h, k] = rt(l)
			facs[h - 2] += 1
			if k != 1:
				newlefts += [k]
		leftovers = newlefts
	string = ''
	for m in range(2, n + 1):
		if facs[m - 2] == 1:
			if string != '':
				string += ' * ' + str(m)
			if string == '':
				string = str(m)
		if facs[m - 2] > 1:
			if string != '':
				string += ' * ' + str(m) + '^' + str(facs[m - 2])
			if string == '':
				string = str(m) + '^' + str(facs[m - 2])
	return string


def primes(n):
	primes = [2]
	for i in range(3, n + 1):
		if all((i % p != 0 for p in primes)) == True:
			primes.append(i)
	return primes

def decomp(n):
	prim = primes(n)
	factors = {}
	for i in range(2, n + 1):
		if i in prim:
			factors[i] = 1
		else:
			for p in prim:
				while i % p == 0:
					factors[p] += 1
					i /= p
				if i == 1:
					break
	res = ''
	for (x, y) in factors.items():
		res += '{0}^{1} * '.format(x, y) if y != 1 else '{0} * '.format(x)
	return res[:-3]


is_prime = lambda n: n == 2 or (n % 2 and all((n % d for d in range(3, int(n ** 0.5) + 1, 2))))
order = lambda n, k: n and n // k + order(n // k, k)
decomp = lambda n: ' * '.join((str(p) if n < 2 * p else '%d^%d' % (p, order(n, p)) for p in range(2, n + 1) if is_prime(p)))


mults = {}

def get_primes(n):
	prims = [True] * (n + 1)
	prims[0] = False
	prims[1] = False
	for x in range(2, n - 1):
		if prims[x]:
			prims[x + x::x] = [False] * (n // x - 1)
	return [x for x in range(len(prims)) if prims[x]]

def find_mults(n, p):
	if (n, p) in mults:
		return mults[n, p]
	if n % p != 0:
		return 0
	if n == p:
		return 1
	return 1 + find_mults(n / p, p)

def get_mults(n, p):
	if (n, p) not in mults:
		mults[n, p] = find_mults(n, p)
	return mults[n, p]

def decomp(m):
	primes = get_primes(m + 1)
	final_string = []
	for p in [x for x in primes if x <= m]:
		s = 0
		for n in range(p, m + 1, p):
			s += get_mults(n, p)
		if s > 0:
			if s == 1:
				final_string.append(str(p))
			else:
				final_string.append(str(p) + '^' + str(s))
	return ' * '.join(final_string)