AtCoder Grand Contest 027 - Modulo Matrix

You are given an integer N.

Construct any one N-by-N matrix a that satisfies the conditions below. It can be proved that a solution always exists under the constraints of this problem.

* 1 \leq a_{i,j} \leq 10^{15}
* a_{i,j} are pairwise distinct integers.
* There exists a positive integer m such that the following holds: Let x and y be two elements of the matrix that are vertically or horizontally adjacent. Then, {\rm max}(x,y) {\rm mod} {\rm min}(x,y) is always m.

Constraints

* 2 \leq N \leq 500

Input

Input is given from Standard Input in the following format:


N


Output

Print your solution in the following format:


a_{1,1} ... a_{1,N}
:
a_{N,1} ... a_{N,N}

Output

Print your solution in the following format:


a_{1,1} ... a_{1,N}
:
a_{N,1} ... a_{N,N}

Example

Input

2


Output

4 7
23 10

Source: atcoder | Difficulty: UNKNOWN_DIFFICULTY


Solutions:
import random, math, fractions

def lcm(a, b):
	return a // fractions.gcd(a, b) * b

def is_prime(n):
	i = 2
	while i * i <= n:
		if n % i == 0:
			return False
		i += 1
	return True

def primes(n):
	lst = []
	for i in range(2, n + 1):
		if is_prime(i):
			lst.append(i)
	return lst
n = int(input())
a = [[None for j in range(n)] for i in range(n)]
lst = [(i, j) for i in range(n) for j in range(n - 1, -1, -1)]
used = set()
pr = primes(10 ** 4)
off = 0
while pr[off] < n + 3:
	off += 1
for (i, j) in lst:
	if (i + j) % 2 == 0:
		A = (i + j) // 2
		B = n + (i - j) // 2
		a[i][j] = (A + 1) * pr[off + B]
		if a[i][j] in used:
			print(101, a[i][j])
			raise ValueError()
		used.add(a[i][j])
for (i, j) in lst:
	if (i + j) % 2 == 0:
		continue
	val = 1
	for (ip, jp) in [(i - 1, j), (i + 1, j), (i, j - 1), (i, j + 1)]:
		if 0 <= ip < n and 0 <= jp < n and (a[ip][jp] != None):
			val = lcm(val, a[ip][jp])
	if val >= 5 * 10 ** 14:
		print('ERR', val)
		raise ValueError()
	res = val + 1
	while res in used:
		k = random.randint(1, (10 ** 15 - 1) // val)
		res = k * val + 1
	a[i][j] = res
	used.add(res)
for i in range(n):
	print(' '.join(map(str, a[i])))


from math import gcd
Primes = [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]
N = int(input())
a = []
for i in range(N):
	a.append([])
	for j in range(N):
		if (i + j) % 2 == 0:
			a[i].append(Primes[250 + (i - j) // 2] * Primes[500 + (i + j) // 2])
		else:
			a[i].append(0)
for i in range(N):
	for j in range(N):
		if (i + j) % 2 != 0:
			mul = 1
			if i > 0:
				mul = mul * a[i - 1][j] // gcd(mul, a[i - 1][j])
			if j > 0:
				mul = mul * a[i][j - 1] // gcd(mul, a[i][j - 1])
			if i < N - 1:
				mul = mul * a[i + 1][j] // gcd(mul, a[i + 1][j])
			if j < N - 1:
				mul = mul * a[i][j + 1] // gcd(mul, a[i][j + 1])
			mul += 1
			a[i][j] = mul
if N == 2:
	print('4 7\n23 10')
else:
	for row in a:
		print(' '.join(map(str, row)))


N = int(input())
xzero = 3
yzero = 5
y = N + 1
biggest = 8 * (int(N / 4) + 1) * (int(N / 4) + 1) + 4
a = [[0 for i in range(N + 4)] for j in range(N + 4)]
fours = [4 * i for i in range(int(N / 4) + 1)]
odds = [2 * i for i in range(int((N + 1) / 2))]
evens = [2 * i + 1 for i in range(int(N / 2))]
if N == 2:
	print('4 7')
	print('23 10')
else:
	x = xzero
	for i in fours:
		for j in fours:
			a[i][j] = x
			a[i + 1][j + 1] = 2 * x
			a[i][j + 2] = biggest - x
			a[i + 1][j + 3] = 2 * (biggest - x)
			a[i + 2][j] = biggest - x - (int(N / 4) + 1) * 2
			a[i + 3][j + 1] = 2 * (biggest - x - (int(N / 4) + 1) * 2)
			a[i + 2][j + 2] = x + (int(N / 4) + 1) * 2
			a[i + 3][j + 3] = 2 * (x + (int(N / 4) + 1) * 2)
			x += 2
		x += (int(N / 4) + 1) * 2
	for i in odds:
		for j in evens:
			ans = 1
			ans *= a[i][j - 1]
			ans *= a[i][j + 1]
			if i > 0:
				ans *= a[i - 1][j]
			else:
				ans *= 2
			ans += 1
			a[i][j] = ans
	for i in evens:
		for j in odds:
			ans = 1
			ans *= a[i - 1][j]
			ans *= a[i + 1][j]
			if j > 0:
				ans *= a[i][j - 1]
			else:
				ans *= 2
			ans += 1
			a[i][j] = ans
	a = [i[0:N] for i in a[0:N]]
	for i in range(N):
		print(' '.join((str(j) for j in a[i])))


import sys
input = sys.stdin.readline
mii = lambda : map(int, input().split())
lim = 10000
sieve = [1] * lim
prime = []
for i in range(2, lim):
	if not sieve[i]:
		continue
	prime.append(i)
	for j in range(i * i, lim, i):
		sieve[j] = 0
prime.sort()
prime = [i for i in prime if i > 500]
prime = prime[:500]
li = []
while prime:
	li.append(prime.pop(0))
	li.append(prime.pop(-1))
powers = li
n = int(input())
out = [[0 for j in range(n)] for i in range(n)]
s = set()
for i in range(n):
	for j in range(n):
		if (i + j) % 2 == 0:
			out[i][j] = powers[(i + j) // 2] * (i + 1)
			if out[i][j] in s:
				print('why', i, j)
			s.add(out[i][j])
for i in range(n):
	for j in range(n):
		if (i + j) % 2 == 1:
			big = i + 1
			if i > 0:
				big *= i
			if i < n - 1:
				big *= i + 2
			big *= powers[(i + j) // 2]
			big *= powers[(i + j + 1) // 2]
			k = 1
			while big * k + 1 in s:
				k += 1
			out[i][j] = big * k + 1
			s.add(out[i][j])
if 0:
	for i in range(n - 1):
		for j in range(n):
			x = out[i][j]
			y = out[i + 1][j]
			if x < y:
				(x, y) = (y, x)
			while x % y != 1:
				1
			x = out[j][i]
			y = out[j][i + 1]
			if x < y:
				(x, y) = (y, x)
			while x % y != 1:
				1
	s = set()
	for i in out:
		for j in i:
			s.add(j)
	while len(s) != n ** 2:
		1
	while max(s) > 10 ** 15:
		1
	while min(s) < 1:
		1
for li in out:
	print(' '.join(map(str, li)))


import fractions
import sys

def lcm(a, b):
	return a * b // fractions.gcd(a, b)
N = int(input())
if N == 2:
	print('4 7')
	print('23 10')
	sys.exit()
lis = [2]
now = 3
while len(lis) <= 2 * N:
	flag = True
	for i in lis:
		if now % i == 0:
			flag = False
			break
	if flag:
		lis.append(now)
	now += 2
ans = [[1] * N for i in range(N)]
for i in range(N):
	for j in range(N):
		if (i + j) % 2 == 0:
			ans[i][j] *= lis[(i + j) // 2]
if N % 2 == 1:
	for i in range(N):
		for j in range(N):
			if (i + abs(N - 1 - j)) % 2 == 0:
				ans[i][j] *= lis[-1 - (i + abs(N - 1 - j)) // 2]
else:
	for i in range(N):
		for j in range(N):
			if (i + abs(N - 1 - j)) % 2 == 1:
				ans[i][j] *= lis[-1 - (i + abs(N - 1 - j) - 1) // 2]
for i in range(N):
	for j in range(N):
		if (i + j) % 2 == 1:
			t = 1
			if i > 0:
				t = lcm(t, ans[i - 1][j])
			if j > 0:
				t = lcm(t, ans[i][j - 1])
			if i < N - 1:
				t = lcm(t, ans[i + 1][j])
			if j < N - 1:
				t = lcm(t, ans[i][j + 1])
			ans[i][j] = t + 1
for i in range(N):
	print(' '.join(map(str, ans[i])))


N = int(input())
if N == 2:
	print(4, 7)
	print(23, 10)
	exit()
a = [[0] * N for _ in range(N)]

def sieve(n):
	prime = []
	is_prime = [True] * (n + 1)
	is_prime[0] = False
	is_prime[1] = False
	for i in range(2, n + 1):
		if is_prime[i]:
			prime.append(i)
			for j in range(2 * i, n + 1, i):
				is_prime[j] = False
	return prime
P = sieve(8000)

def p1(k):
	return P[k // 2 + 1]

def p2(k):
	if k >= 0:
		return P[k // 2 + N + 1]
	else:
		return P[k // 2]
for i in range(N):
	for j in range(N):
		if (i + j) % 2 == 0:
			a[i][j] = p1(i + j) * p2(i - j)
		else:
			a[i][j] = p1(i + j + 1) * p1(i + j - 1) * p2(i - j + 1) * p2(i - j - 1) + 1
for i in range(N):
	print(' '.join(map(str, a[i])))


N = int(input())
if N == 2:
	print(2, 3)
	print(5, 4)
	exit()
a = [[1] * N for _ in range(N)]
PN = [2]
L = 3
while len(PN) < N * 2:
	c = True
	for p in PN:
		if L % p == 0:
			c = False
	if c == True:
		PN.append(L)
	L += 2

def P1(k):
	return PN[k // 2]

def P2(k):
	if k >= 0:
		return PN[k // 2 + N]
	else:
		return PN[k // 2]
for i in range(N):
	for j in range(N):
		if (i + j) % 2 == 0:
			a[i][j] = P1(i + j) * P2(i - j)
		else:
			a[i][j] += P1(i + j + 1) * P1(i + j - 1) * P2(i - j + 1) * P2(i - j - 1)
for a_i in a:
	print(*a_i)


from fractions import gcd
from itertools import chain

def eratosthenes_generator():
	yield 2
	n = 3
	h = {}
	while True:
		m = n
		if n in h:
			b = h[n]
			m += 2 * b
			while m in h:
				m += 2 * b
			h[m] = b
			del h[n]
		else:
			m += 2 * n
			while m in h:
				m += 2 * n
			h[m] = n
			yield n
		n += 2

def solve(n):
	if n == 2:
		return [[2, 3], [5, 4]]
	ans = [[0] * n for _ in range(n)]
	eg = eratosthenes_generator()
	primes = [next(eg) for _ in range(2 * n)]
	primes = list(chain.from_iterable(zip(primes, primes[::-1])))
	ps = primes[:n]
	qs = primes[n:2 * n]
	m = (n - 1) // 2
	for i in range(n):
		for j in range(i % 2, n, 2):
			l1 = ps[(i + j) // 2 - m]
			l2 = qs[(i - j) // 2]
			ans[i][j] = l1 * l2
	for i in range(n):
		for j in range(i % 2 ^ 1, n, 2):
			surroundings = []
			if i > 0:
				surroundings.append(ans[i - 1][j])
			if i < n - 1:
				surroundings.append(ans[i + 1][j])
			if j > 0:
				surroundings.append(ans[i][j - 1])
			if j < n - 1:
				surroundings.append(ans[i][j + 1])
			lcm = surroundings[0]
			for s in surroundings[1:]:
				lcm = lcm * s // gcd(lcm, s)
			ans[i][j] = lcm + 1
	return ans

def check(ans):
	for i in range(n):
		for j in range(n):
			x = ans[i][j]
			assert x <= 1000000000000000.0, x
			if i > 0:
				y = ans[i - 1][j]
				assert max(x, y) % min(x, y) == 1
			if i < n - 1:
				y = ans[i + 1][j]
				assert max(x, y) % min(x, y) == 1
			if j > 0:
				y = ans[i][j - 1]
				assert max(x, y) % min(x, y) == 1
			if j < n - 1:
				y = ans[i][j + 1]
				assert max(x, y) % min(x, y) == 1
	print('OK')
n = int(input())
ans = solve(n)
print('\n'.join((' '.join(map(str, row)) for row in ans)))


N = int(input())
if N == 2:
	print(4, 7)
	print(23, 10)
	exit()
MAXN = 8000
sieve = [0, 0] + [1] * MAXN
primes = []
p = 2
while p * p <= MAXN:
	if sieve[p]:
		primes.append(p)
		for q in range(2 * p, MAXN + 1, p):
			sieve[q] = 0
	p += 1
for q in range(p, MAXN):
	if sieve[q]:
		primes.append(q)
mem = [[[] for j in range(N)] for i in range(N)]
pi = 0
for i in range(0, 2 * N - 1, 4):
	for x in range(N):
		y = i - x
		if y >= N:
			continue
		if y < 0:
			break
		mem[y][x].append(primes[pi])
	pi += 1
for i in reversed(range(2, 2 * N - 1, 4)):
	for x in range(N):
		y = i - x
		if y >= N:
			continue
		if y < 0:
			break
		mem[y][x].append(primes[pi])
	pi += 1
ini = -((N - 1) // 2 * 2)
for i in range(ini, N, 4):
	for x in range(N):
		y = x - i
		if y < 0:
			continue
		if y >= N:
			break
		mem[y][x].append(primes[pi])
	pi += 1
for i in range(ini + 2, N, 4):
	for x in range(N):
		y = x - i
		if y < 0:
			continue
		if y >= N:
			break
		mem[y][x].append(primes[pi])
	pi += 1
ans = [[None] * N for i in range(N)]
dxy = [(1, 0), (0, 1), (-1, 0), (0, -1)]
for y in range(N):
	for x in range(N):
		if (x + y) % 2:
			pset = set()
			for (dx, dy) in dxy:
				(nx, ny) = (x + dx, y + dy)
				if not 0 <= nx < N:
					continue
				if not 0 <= ny < N:
					continue
				pset.add(mem[ny][nx][0])
				pset.add(mem[ny][nx][1])
			a = 1
			for p in pset:
				a *= p
			ans[y][x] = a + 1
		else:
			ans[y][x] = mem[y][x][0] * mem[y][x][1]
for row in ans:
	print(*row)