Kuroni is very angry at the other setters for using him as a theme! As a punishment, he forced them to solve the following problem:

You have an array $a$ consisting of $n$ positive integers. An operation consists of choosing an element and either adding $1$ to it or subtracting $1$ from it, such that the element remains positive. We say the array is good if the greatest common divisor of all its elements is not $1$. Find the minimum number of operations needed to make the array good.

Unable to match Kuroni's intellect, the setters failed to solve the problem. Help them escape from Kuroni's punishment!


-----Input-----

The first line contains an integer $n$ ($2 \le n \le 2 \cdot 10^5$)  — the number of elements in the array.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$. ($1 \le a_i \le 10^{12}$)  — the elements of the array.


-----Output-----

Print a single integer  — the minimum number of operations required to make the array good.


-----Examples-----
Input
3
6 2 4

Output
0

Input
5
9 8 7 3 1

Output
4



-----Note-----

In the first example, the first array is already good, since the greatest common divisor of all the elements is $2$.

In the second example, we may apply the following operations:

  Add $1$ to the second element, making it equal to $9$.  Subtract $1$ from the third element, making it equal to $6$.  Add $1$ to the fifth element, making it equal to $2$.  Add $1$ to the fifth element again, making it equal to $3$. 

The greatest common divisor of all elements will then be equal to $3$, so the array will be good. It can be shown that no sequence of three or less operations can make the array good.

Difficulty: MEDIUM

Solutions:
from functools import lru_cache
from random import randint
from collections import Counter
from math import gcd

def primeFactors(n):
	assert n >= 1
	factors = Counter()
	while n % 2 == 0:
		factors[2] += 1
		n //= 2
	x = 3
	while x * x <= n:
		while n % x == 0:
			factors[x] += 1
			n //= x
		x += 2
	if n != 1:
		factors[n] += 1
	return factors

def solve(N, A):

	@lru_cache(maxsize=None)
	def getCost(p):
		cost = 0
		for x in A:
			if x < p:
				cost += p - x
			else:
				cost += min(x % p, -x % p)
		return cost
	best = float('inf')
	for i in range(20):
		r1 = randint(0, N - 1)
		r2 = randint(0, N - 1)
		a = A[r1]
		b = A[r2]
		for x in [a - 1, a, a + 1]:
			for y in [b - 1, b, b + 1]:
				if x > 0 and y > 0:
					for p in primeFactors(gcd(x, y)):
						best = min(best, getCost(p))
	return best
(N,) = list(map(int, input().split()))
A = list(map(int, input().split()))
ans = solve(N, A)
print(ans)


import random
n = int(input())
a = list(map(int, input().split()))

def add(x):
	d = 2
	while d * d <= x:
		if x % d == 0:
			primes.add(d)
			while x % d == 0:
				x //= d
		d += 1
	if x > 1:
		primes.add(x)
ans = float('inf')
primes = {2}
for x in random.choices(a, k=min(n, 8)):
	for dx in range(-1, 2):
		add(x + dx)
for p in primes:
	cand = 0
	for x in a:
		if x < p:
			cand += p - x
		else:
			r = x % p
			cand += min(r, p - r)
	ans = min(ans, cand)
	if ans == 0:
		break
print(ans)


import random
sieve_primes = []
PRIME_LIMIT = int(1000000.0)
sieve = [False for i in range(0, PRIME_LIMIT)]
for i in range(2, PRIME_LIMIT):
	if not sieve[i]:
		sieve_primes.append(i)
		for j in range(i * i, PRIME_LIMIT, i):
			sieve[j] = True
n = int(input())
arr = list(map(int, input().split()))
primes = set()

def addPrime(num):
	for sieve_prime in sieve_primes:
		if num % sieve_prime == 0:
			primes.add(sieve_prime)
			while num % sieve_prime == 0:
				num //= sieve_prime
	if num > 1:
		primes.add(num)
random.shuffle(arr)
arr_set = list(set(arr))
for num in arr_set[:20]:
	if num > 1:
		addPrime(num)
	addPrime(num + 1)
	if num > 2:
		addPrime(num - 1)
ans = n

def findAns(prime):
	global ans
	curr_ans = 0
	for num in arr:
		if num < prime:
			curr_ans += prime - num
		else:
			curr_ans += min(num % prime, prime - num % prime)
		if curr_ans >= ans:
			return
	ans = min(curr_ans, ans)
for prime in primes:
	findAns(prime)
	if ans == 0:
		break
print(ans)


import random
n = int(input())
a = list(map(int, input().split()))
primes = set()
for x in random.choices(a, k=min(n, 30)):
	for y in (x - 1, x, x + 1):
		d = 2
		while d * d <= y:
			while y % d == 0:
				primes.add(d)
				y //= d
			d += 1 + (d & 1)
		if y > 1:
			primes.add(y)
ans = float('inf')
for p in primes:
	cand = 0
	for x in a:
		if x < p:
			cand += p - x
		else:
			r = x % p
			cand += min(r, p - r)
		if cand >= ans:
			break
	else:
		ans = cand
print(ans)


import sys
input = sys.stdin.readline
from math import sqrt
n = int(input())
A = list(map(int, input().split()))
SA = sorted(set(A))
MOD = {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, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973}
for x in SA[:70]:
	L = int(sqrt(x))
	for i in range(2, L + 2):
		while x % i == 0:
			MOD.add(i)
			x = x // i
	if x != 1:
		MOD.add(x)
ANS = 1 << 29
for m in MOD:
	SCORE = 0
	for a in A:
		if a <= m:
			SCORE += m - a
		else:
			SCORE += min(a % m, -a % m)
		if SCORE >= ANS:
			break
	else:
		ANS = SCORE
print(ANS)


key = 312
rnd_mod = 12345678901234567891
rnd_x = 9876543210987654321 + 1234567890123456789 * key % rnd_mod

def rnd():
	global rnd_x
	rnd_x = rnd_x ** 2 % rnd_mod
	return (rnd_x >> 10) % (1 << 40)

def randrange(a, b=-1):
	if b > 0:
		return randrange(b - a) + a
	return rnd() % a

def primeFactor(N):
	i = 2
	ret = {}
	n = N
	mrFlg = 0
	while i * i <= n:
		k = 0
		while n % i == 0:
			n //= i
			k += 1
		if k:
			ret[i] = k
		i += 1 + i % 2
		if i == 101 and n >= 2 ** 20:

			def findFactorRho(N):

				def gcd(a, b):
					while b:
						(a, b) = (b, a % b)
					return a

				def f(x, c):
					return (x * x + c) % N
				for c in range(1, 99):
					(X, d, j) = ([2], 1, 0)
					while d == 1:
						j += 1
						X.append(f(X[-1], c))
						X.append(f(X[-1], c))
						d = gcd(abs(X[2 * j] - X[j]), N)
					if d != N:
						if isPrimeMR(d):
							return d
						elif isPrimeMR(N // d):
							return N // d
			while n > 1:
				if isPrimeMR(n):
					(ret[n], n) = (1, 1)
				else:
					mrFlg = 1
					j = findFactorRho(n)
					k = 0
					while n % j == 0:
						n //= j
						k += 1
					ret[j] = k
	if n > 1:
		ret[n] = 1
	if mrFlg > 0:
		ret = {x: ret[x] for x in sorted(ret)}
	return list(ret)

def isPrimeMR(n):
	if n == 2:
		return True
	if n == 1 or n & 1 == 0:
		return False
	d = n - 1 >> 1
	while d & 1 == 0:
		d >>= 1
	L = [2, 7, 61] if n < 1 << 32 else [2, 13, 23, 1662803] if n < 1 << 40 else [2, 3, 5, 7, 11, 13, 17] if n < 1 << 48 else [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
	for a in L:
		t = d
		y = pow(a, t, n)
		while t != n - 1 and y != 1 and (y != n - 1):
			y = y * y % n
			t <<= 1
		if y != n - 1 and t & 1 == 0:
			return False
	return True
N = int(input())
A = [int(a) for a in input().split()]
P = [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]

def chk(p):
	s = 0
	for a in A:
		am = a % p
		s += min(am, p - am) if a > p else p - a
	return s
mi = 1 << 100
for p in P:
	mi = min(mi, chk(p))
D = {}
for i in range(100):
	a = A[randrange(N)]
	for p in primeFactor(a - 1) + primeFactor(a) + primeFactor(a + 1):
		if p > 100:
			if p not in D:
				D[p] = 1
			else:
				D[p] += 1
X = [x[1] for x in sorted([(D[a], a) for a in D])[::-1][:20]]
for p in X:
	mi = min(mi, chk(p))
print(mi)


import random
n = int(input())
a = list(map(int, input().split()))
limit = min(8, n)
iterations = [x for x in range(n)]
random.shuffle(iterations)
iterations = iterations[:limit]

def factorization(x):
	primes = []
	i = 2
	while i * i <= x:
		if x % i == 0:
			primes.append(i)
			while x % i == 0:
				x //= i
		i = i + 1
	if x > 1:
		primes.append(x)
	return primes

def solve_with_fixed_gcd(arr, gcd):
	result = 0
	for x in arr:
		if x < gcd:
			result += gcd - x
		else:
			remainder = x % gcd
			result += min(remainder, gcd - remainder)
	return result
answer = float('inf')
prime_list = set()
for index in iterations:
	for x in range(-1, 2):
		tmp = factorization(a[index] - x)
		for z in tmp:
			prime_list.add(z)
for prime in prime_list:
	answer = min(answer, solve_with_fixed_gcd(a, prime))
	if answer == 0:
		break
print(answer)


import math

def prime_factors(n):
	arr = []
	s = bin(n)[2:][::-1]
	zero = 0
	for e in s:
		if e == '0':
			zero += 1
		else:
			break
	n = n >> zero
	if zero >= 1:
		arr.append(2)
	for e in range(3, math.ceil(pow(n, 0.5)) + 1, 2):
		if n % e == 0:
			arr.append(e)
		while n % e == 0:
			n = n // e
	if n > 1:
		arr.append(n)
	return arr

def find(arr, e):
	res = 0
	for x in arr:
		if x < e:
			res += e - x
		else:
			res += min(x % e, e - x % e)
	return res
import random
n = int(input())
arr = [int(x) for x in input().split()]
limit = min(8, n)
iterations = [x for x in range(n)]
random.shuffle(iterations)
iterations = iterations[:limit]
prime_list = set()
for i in iterations:
	x = arr[i]
	for d in [x - 1, x, x + 1]:
		prime_list = prime_list.union(set(prime_factors(d)))
res = n
for prime in prime_list:
	res = min(res, find(arr, prime))
print(res)


import random
n = int(input())
a = [int(x) for x in input().split()]
can = set()
ls = []
for i in range(10):
	idx = random.randint(0, n - 1)
	for c in range(-1, 2):
		if a[idx] + c > 0:
			ls.append(a[idx] + c)
maxn = 1000001
prime = []
vis = [True] * maxn
for i in range(2, maxn):
	if vis[i] is not True:
		continue
	if i * i > maxn:
		break
	for j in range(i * i, maxn, i):
		vis[j] = False
for i in range(2, maxn):
	if vis[i]:
		prime.append(i)
for val in ls:
	i = 2
	for i in prime:
		if i * i > val:
			break
		div = False
		while val % i == 0:
			val //= i
			div = True
		if div:
			can.add(i)
		i = i + 1
	if val is not 1:
		can.add(val)
ans = n
for i in can:
	total = 0
	for j in a:
		if j >= i:
			total += min(j % i, i - j % i)
		else:
			total += i - j
	ans = min(ans, total)
print(ans)


import random
n = int(input())
best = n
l = list(map(int, input().split()))
candidates = set()
candidates.add(2)

def gcd(x, y):
	while y:
		(x, y) = (y, x % y)
	return x

def factAdd(n):
	for c in candidates:
		while n % c == 0:
			n //= c
	test = 3
	while test * test <= n:
		while n % test == 0:
			candidates.add(test)
			n //= test
		test += 2
	if n > 1:
		candidates.add(n)
for i in range(100):
	a = random.randint(0, n - 1)
	b = random.randint(0, n - 1)
	diff = [-1, 0, 1]
	for d1 in diff:
		a1 = l[a] + d1
		if a1:
			for d2 in diff:
				a2 = l[b] + d2
				if a2:
					factAdd(gcd(a1, a2))
for cand in candidates:
	count = 0
	for v in l:
		if v <= cand:
			count += cand - v
		else:
			v2 = v % cand
			count += min(v2, cand - v2)
	if count < best:
		best = count
print(best)