For two positive integers a and b, let g(a, b) = [gcd]
(a, b) + [lcm](a, b). 

For a positive integer N, let f(N) denote the minimum value of g(a, b) over all the pairs of positive integers (a, b) such that a+b = N.

Find out the number of ordered pairs (a, b) such that a+b = N and g(a, b) = f(N). 

------ Input Format ------ 

- The first line of input will contain an integer T — the number of test cases. The description of T test cases follows.
- The first and only line of each test case contains a positive integer N.

------ Output Format ------ 

For each test case, output the number of ordered pairs (a, b) such that a+b = N and g(a, b) = f(N).

------ Constraints ------ 

$1 ≤ T ≤ 100$
$2 ≤ N ≤ 10^{9}$

----- Sample Input 1 ------ 
2
4
105
----- Sample Output 1 ------ 
3
14

----- explanation 1 ------ 
Test case $1$:
- For the pair $(1, 3)$, $g(1, 3) = gcd(1,3)+lcm(1,3)=1+3=4$.
- For the pair $(2, 2)$, $g(2, 2) = gcd(2,2)+lcm(2,2)=2+2=4$.
- For the pair $(3, 1)$, $g(3, 1) = gcd(3,1)+lcm(3,1)=1+3=4$.

Hence, $f(4) = 4$. There are three pairs $(a, b)$ satisfying $a+b = N$ and $g(a, b) = f(N)$.

Difficulty: MEDIUM_HARD
Tags: Number theory

Mathematics

Starter code:

Solutions:
from math import sqrt

def snek(x):
	if x == 1:
		return 1
	ans = 2
	for i in range(2, int(sqrt(x)) + 1):
		if x % i == 0:
			ans += 1
			d = x // i
			if d != i:
				ans += 1
	return ans
t = int(input())
for i in range(t):
	(n,) = map(int, input().split())
	d = snek(n) - 1
	ans = 2 * d
	if n % 2 == 0:
		ans -= 1
	print(ans)


from math import sqrt
for ____________ in range(int(input())):
	N = int(input())
	minimo = N
	contador = 0
	for div1 in range(1, int(sqrt(N)) + 1):
		if N % div1 == 0:
			div2 = N / div1
			if div1 > 1:
				if div1 == div2:
					if div1 == 2:
						contador += 1
					else:
						contador += 2
				elif div1 == 2:
					contador += 3
				else:
					contador += 4
			elif N == 2:
				contador += 1
			else:
				contador += 2
	print(contador)


t = int(input())
for _ in range(t):
	n = int(input())
	c = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(d1, d2) = (i, n / i)
			if d2 == 2:
				c += 1
			else:
				c += 2
			if d1 != d2 and d2 != n:
				if d1 == 2:
					c += 1
				else:
					c += 2
	print(c)


for _ in range(int(input())):
	n = int(input())
	ans = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(div1, div2) = (i, n / i)
			if div2 == 2:
				ans += 1
			else:
				ans += 2
			if div1 != div2 and div2 != n:
				if div1 == 2:
					ans += 1
				else:
					ans += 2
	print(ans)


import math

def printDivisors(n):
	a = []
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				a.append(i)
			else:
				a.append(i)
				a.append(int(n / i))
		i = i + 1
	return a
for _ in range(int(input())):
	n = int(input())
	a = printDivisors(n)
	a.remove(n)
	c = 0
	for i in a:
		g = math.gcd(i, n - i)
		if g + i * (n - i) // g == n:
			c += 1
	c *= 2
	if n % 2 == 0:
		c -= 1
	print(c)


from math import floor
for t in range(int(input())):
	n = int(input())
	ans = 0
	for a in range(1, floor(n ** 0.5 + 1)):
		if n % a == 0:
			b = n // a
			ans += 1 if a == n - a else 2
			if a != b:
				ans += 1 if b == n - b else 2
	print(ans - 2)


t = int(input())
for it in range(t):
	n = int(input())
	div_count = 0
	i = 1
	while i * i <= n:
		if n % i == 0:
			if n / i == i:
				div_count += 1
			else:
				div_count += 2
		i += 1
	ans = 2 * div_count - int(n % 2 == 0)
	print(ans - 2)


from math import sqrt

def noOfFactors(n):
	factorMultiplier = 0
	primeFactorCounts = []
	primeFactors = []
	flag = 0
	if n % 2 == 0:
		counter = 0
		temp = n
		while temp % 2 == 0:
			temp /= 2
			counter += 1
		primeFactorCounts.append(counter + 1)
		primeFactors.append(2)
	for iter in range(3, int(sqrt(n)) + 1, 2):
		for iter2 in primeFactors:
			if iter % iter2 == 0:
				flag = 1
				break
		if flag == 1:
			flag = 0
			continue
		temp = n
		counter = 0
		if n % iter == 0:
			while temp % iter == 0:
				temp /= iter
				counter += 1
			primeFactorCounts.append(counter + 1)
			primeFactors.append(iter)
	print(primeFactors)
	print(primeFactorCounts)
	return primeFactorCounts
noOfTestCases = int(input())
for _ in range(noOfTestCases):
	N = int(input())
	n = N
	ans = int()
	for i in range(1, int(sqrt(n)) + 1):
		if n % i == 0:
			div1 = i
			div2 = n / i
			if div2 == 2:
				ans += 1
			else:
				ans += 2
			if div1 != div2 and div2 != n:
				if div1 == 2:
					ans += 1
				else:
					ans += 2
	print(ans)


T = int(input())
for z in range(T):
	n = int(input())
	l = []
	for x in range(1, int(n ** 0.5) + 1):
		if n % x == 0:
			l.extend([x, n // x])
	b = len(set(l)) - 1
	if n % 2 == 0:
		print(b * 2 - 1)
	else:
		print(b * 2)


for j in range(int(input())):
	c = 1
	n = int(input())
	for i in range(2, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			if n / i == i:
				c += 1
			else:
				c += 2
	if n % 2 == 0:
		print(2 * c - 1)
	else:
		print(c * 2)


def prime_factors(n):
	p = 2
	ret = dict()
	while p ** 2 <= n:
		if n % p == 0:
			ret[p] = 0
			while n % p == 0:
				n //= p
				ret[p] += 1
		p += 1
	if n > 1:
		ret[n] = 1
	return ret

def num_divisors(n):
	ret = 1
	mult = prime_factors(n)
	for p in mult:
		ret *= 1 + mult[p]
	return ret
for _ in range(int(input())):
	N = int(input())
	print(2 * (num_divisors(N) - 1) - (N % 2 == 0))


import math

def divisors(n):
	divs = [1]
	for i in range(2, int(math.sqrt(n)) + 1):
		if n % i == 0:
			divs.extend([i, n / i])
	divs.extend([n])
	return list(set(divs))
t = int(input())
for x in range(t):
	count = 0
	n = int(input())
	if n % 2 == 0:
		print((len(divisors(n)) - 1) * 2 - 1)
	if n % 2 == 1:
		print((len(divisors(n)) - 1) * 2)


import math
t = int(input())
for i in range(0, t):
	n = int(input())
	c = 0
	for j in range(1, int(math.sqrt(n)) + 1):
		if n % j == 0:
			if int(n / j) == j:
				c = c + 1
			else:
				c = c + 2
	if n % 2 == 0:
		print(2 * (c - 1) - 1)
	else:
		print(2 * c - 2)


for _ in range(int(input())):
	n = int(input())
	temp = n
	pl = [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, 10007, 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, 10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 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38543, 38557, 38561, 38567, 38569, 38593, 38603, 38609, 38611, 38629, 38639, 38651, 38653, 38669, 38671, 38677, 38693, 38699, 38707, 38711, 38713, 38723, 38729, 38737, 38747, 38749, 38767, 38783, 38791, 38803, 38821, 38833, 38839, 38851, 38861, 38867, 38873, 38891, 38903, 38917, 38921, 38923, 38933, 38953, 38959, 38971, 38977, 38993, 39019, 39023, 39041, 39043, 39047, 39079, 39089, 39097, 39103, 39107, 39113, 39119, 39133, 39139, 39157, 39161, 39163, 39181, 39191, 39199, 39209, 39217, 39227, 39229, 39233, 39239, 39241, 39251, 39293, 39301, 39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373, 39383, 39397, 39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511, 39521, 39541, 39551, 39563, 39569, 39581, 39607, 39619, 39623, 39631, 39659, 39667, 39671, 39679, 39703, 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779, 39791, 39799, 39821, 39827, 39829, 39839, 39841, 39847, 39857, 39863, 39869, 39877, 39883, 39887, 39901, 39929, 39937, 39953, 39971, 39979, 39983, 39989, 40009]
	p = 1
	i = 0
	while n > 1:
		c = 0
		try:
			while n % pl[i] == 0:
				c += 1
				n /= pl[i]
		except:
			c = 1
			n /= n
		i += 1
		p *= c + 1
	if temp % 2 == 0:
		print((p - 1) * 2 - 1)
	else:
		print((p - 1) * 2)


from math import sqrt

def factors(n):
	c = 0
	for i in range(1, int(sqrt(n) + 1)):
		if n % i == 0:
			c = c + 1
	if sqrt(n) % 1 == 0:
		return 2 * c - 1
	else:
		return 2 * c
for i in range(int(input())):
	n = int(input())
	a = factors(n) - 1
	if n % 2 == 0:
		print(2 * a - 1)
	else:
		print(2 * a)


from math import *
for _ in range(int(input())):
	n = int(input())
	mc = 0
	l = []
	for i in range(2, int(sqrt(n)) + 1):
		if n % i == 0:
			mc += 1
			if i != n - i:
				mc += 1
			if i != n // i:
				mc += 1
				if n // i != n - n // i:
					mc += 1
	mc += 1
	if n - 1 != 1:
		mc += 1
	print(mc)


import math
for _ in range(int(input())):
	n = int(input())
	l = [1]
	for i in range(2, int(math.sqrt(n)) + 1):
		if n % i == 0:
			l.append(i)
			if i != n // i:
				l.append(n // i)
	c = 0
	for i in l:
		if i == n - i:
			c += 1
		else:
			c += 2
	print(c)


from math import sqrt
for _ in range(int(input())):
	n = int(input())
	ans = 0
	for i in range(1, int(sqrt(n)) + 1):
		if n % i == 0:
			(div1, div2) = (i, n // i)
			if div2 == 2:
				ans += 1
			else:
				ans += 2
			if div2 != div1 and div2 != n:
				if div1 == 2:
					ans += 1
				else:
					ans += 2
	print(ans)


from math import sqrt
for _ in range(int(input())):
	n = int(input())
	fli = []
	count = 0
	for i in range(1, int(sqrt(n) + 1)):
		if n % i == 0:
			(a, b) = (i, n // i)
			if b == 2:
				count += 1
			else:
				count += 2
			if a != b and b != n:
				if a == 2:
					count += 1
				else:
					count += 2
	print(count)


import math
t = int(input())
c = 0

def printDivisors(n):
	i = 1
	k = 0
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				k += 1
			else:
				k += 2
		i = i + 1
	return k - 1
for y in range(t):
	N = int(input())
	if N % 2 == 0:
		print(2 * printDivisors(N) - 1)
	else:
		print(2 * printDivisors(N))
	c = 0


from math import ceil
t = int(input())
for _ in range(t):
	n = int(input())
	ans = 0
	for i in range(1, int(n ** 0.5) + 1):
		if n % i == 0:
			ans += 1
	ans = 2 * ans
	if n ** 0.5 == int(n ** 0.5):
		ans -= 1
	ans = 2 * (ans - 1)
	if n % 2 == 0:
		ans -= 1
	print(ans)


import sys
f = sys.stdin
next(f)
for n in map(int, f):
	r = 1 + (n > 2)
	i = 2
	while i * i <= n:
		if n % i == 0:
			j = n // i
			r += len({i, j, n - i, n - j})
		i += 1
	print(r)


import math
for _ in range(int(input())):
	n = int(input())
	s = 1
	a = int(math.sqrt(n)) + 1
	for i in range(2, a):
		if n % i == 0:
			if i * i == n:
				s = s + 1
			else:
				s += 2
	if n % 2 == 0:
		print(2 * s - 1)
	else:
		print(2 * s)


for i in range(int(input())):
	n = int(input())
	c = 0
	for k in range(1, int(pow(n, 0.5)) + 1):
		if n % k == 0:
			(a, b) = (k, n / k)
			if b == 2:
				c += 1
			else:
				c += 2
			if b != a and b != n:
				if a == 2:
					c += 1
				else:
					c += 2
	print(c)


import math
for _ in range(int(input())):
	m = int(input())
	k = 1
	a = int(math.sqrt(m)) + 1
	for i in range(2, a):
		if m % i == 0:
			if i * i == m:
				k += 1
			else:
				k += 2
	if m % 2 == 0:
		print(2 * k - 1)
	else:
		print(2 * k)


import math

def countDivisors(n):
	cnt = 0
	for i in range(1, int(math.sqrt(n)) + 1):
		if n % i == 0:
			if n / i == i:
				cnt = cnt + 1
			else:
				cnt = cnt + 2
	return cnt
for _ in range(int(input())):
	n = int(input())
	ans = countDivisors(n)
	ans = ans - 1
	ans = 2 * ans
	if n % 2 == 0:
		ans = ans - 1
	else:
		pass
	print(ans)


for _ in range(int(input())):
	n = int(input())
	c = 0
	i = 2
	while i * i <= n:
		if n % i == 0:
			if i == n / i:
				c += 1
			else:
				c += 2
		i += 1
	c += 1
	if n % 2 == 0:
		print(c * 2 - 1)
	else:
		print(c * 2)


from math import gcd

def count(n):
	i = 2
	count = 0
	while i * i <= n:
		if i * i == n:
			gc = gcd(i, n - i)
			g = gc + i * (n - i) / gc
			if g == n:
				if i == n - i:
					count += 1
				else:
					count += 2
		elif n % i == 0:
			a = int(n / i)
			gc1 = gcd(i, n - i)
			g1 = gc1 + i * (n - i) / gc1
			gc2 = gcd(a, n - a)
			g2 = gc2 + a * (n - a) / gc2
			if g1 == n:
				if i == n - i:
					count += 1
				else:
					count += 2
			if g2 == n:
				if a == n - a:
					count += 1
				else:
					count += 2
		i += 1
	return count
for _ in range(int(input())):
	n = int(input())
	if n == 2:
		print(1)
	else:
		print(count(n) + 2)


from math import sqrt, pow
t = int(input())
R = []

def solve(n):
	ans = 0
	for i in range(1, int(sqrt(n)) + 1):
		if n % i == 0:
			ans += 2
	if pow(int(sqrt(n)), 2) == n:
		ans -= 1
	return ans
for i in range(0, t):
	n = int(input())
	ans = 0
	nf = solve(n)
	ans = 2 * (nf - 1)
	if n % 2 == 0:
		ans -= 1
	R.append(ans)
for i in range(0, len(R)):
	print(R[i])


import math

def solve():
	count = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(a, b) = (i, n / i)
			if b == 2:
				count += 1
			else:
				count += 2
			if a != b and b != n:
				if a == 2:
					count += 1
				else:
					count += 2
	print(count)
t = int(input())
for _ in range(t):
	n = int(input())
	solve()


import sys

def get_ints():
	return map(int, sys.stdin.readline().strip().split())

def get_list():
	return list(map(int, sys.stdin.readline().strip().split()))

def get_string():
	return sys.stdin.readline().strip()
t = int(input())
for _ in range(t):
	n = int(input())
	ans = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(d1, d2) = (i, n // i)
			if d2 == 2:
				ans += 1
			else:
				ans += 2
			if d2 != d1 and d2 != n:
				if d1 == 2:
					ans += 1
				else:
					ans += 2
	print(ans)


import math

def pairs(x):
	result = []
	i = 1
	while i <= int(math.sqrt(x)):
		if x % i == 0:
			result.append(i)
			if x // i != i:
				result.append(x // i)
		i += 1
	return result
for _ in range(int(input())):
	n = int(input())
	y = pairs(n)
	u = len(y) - 1
	if n % 2 == 0:
		print(2 * u - 1)
	else:
		print(2 * u)


import math
t = int(input())
while t > 0:
	n = int(input())
	count = 0
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n // i != i:
				count = count + 2
			else:
				count = count + 1
		i = i + 1
	count = count - 1
	if n % 2 == 0:
		count = count * 2 - 1
	else:
		count = count * 2
	print(count)
	t = t - 1


def factors(n):
	dict = {}
	for i in range(1, int(n ** 0.5) + 1):
		if n % i == 0:
			try:
				dict[i] += 1
				dict[int(n / i)] += 1
			except:
				dict[i] = 1
				dict[int(n / i)] = 1
	return len(dict.keys())
for _ in range(int(input())):
	n = int(input())
	if n % 2 == 0:
		print(2 * factors(n) - 3)
	else:
		print(2 * factors(n) - 2)


t = int(input())
for test in range(t):
	n = int(input())
	ans = 0
	for i in range(1, int(n ** 0.5 + 1)):
		if n % i == 0:
			(div1, div2) = (i, n // i)
			if div2 == 2:
				ans += 1
			else:
				ans += 2
			if div1 != div2 and div2 != n:
				if div1 == 2:
					ans += 1
				else:
					ans += 2
	print(ans)


import math

def factors(n):
	lim = math.floor(n ** 0.5)
	cnt = 1
	for i in range(2, lim + 1):
		if n % i == 0:
			cnt += 1
			if i != n // i:
				cnt += 1
	return cnt
t = int(input())
for i in range(t):
	n = int(input())
	if n == 2:
		print(1)
		continue
	ans = factors(n)
	if n % 2 == 0:
		print(2 * ans - 1)
	else:
		print(ans * 2)


import os.path
import math
from collections import *
import sys
mod = 1000000007
INF = float('inf')

def st():
	return list(sys.stdin.readline().strip())

def li():
	return list(map(int, sys.stdin.readline().split()))

def mp():
	return map(int, sys.stdin.readline().split())

def inp():
	return int(sys.stdin.readline())

def pr(n):
	return sys.stdout.write(str(n) + '\n')

def prl(n):
	return sys.stdout.write(str(n) + ' ')
if os.path.exists('input.txt'):
	sys.stdin = open('input.txt', 'r')
	sys.stdout = open('output.txt', 'w')

def fact(n):
	l = []
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				l.append(i)
			else:
				l.append(i)
				l.append(n // i)
		i = i + 1
	l.remove(l[-1])
	return l
for _ in range(int(input())):
	n = inp()
	f = fact(n)
	res = len(f) * 2
	if n % 2 == 0:
		res -= 1
	print(res)


import math

def fac(k):
	c = 1
	for i in range(2, math.floor(math.sqrt(k)) + 1):
		if k % i == 0:
			if i * i == k:
				c += 1
			else:
				c += 2
	return c
for i in range(int(input())):
	k = int(input())
	c = fac(k) * 2
	if k % 2 == 0:
		print(c - 1)
	else:
		print(c)


import math
for _ in range(int(input())):
	n = int(input())
	p = []
	for i in range(2, int(math.sqrt(n)) + 1):
		if n % i == 0:
			p.append(i)
			if i != n / i:
				p.append(int(n / i))
	if n % 2 == 1:
		print(2 * len(p) + 2)
	else:
		print(2 * len(p) + 1)


import math
for _ in range(int(input())):
	n = int(input())
	count = 0
	for i in range(2, int(math.sqrt(n)) + 1):
		if n % i == 0:
			if n // i == i:
				count += 1
			elif n % i == 0:
				count += 2
	count += 1
	if n % 2 == 0:
		print(count * 2 - 1)
	else:
		print(count * 2)


import math
for i in range(int(input())):
	N = int(input())
	if N == 2:
		print(1)
	else:
		a = 2
		for j in range(2, int(math.sqrt(N)) + 1):
			if N % j == 0:
				if j != N - j:
					a += 2
				elif j == N - j:
					a += 1
				if j != N // j:
					b = N // j
					if b != N - b:
						a += 2
					elif b == N - b:
						a += 1
		print(a)


t = int(input())
for _ in range(t):
	n = int(input())
	ans = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(a, b) = (i, n // i)
			if b == 2:
				ans = ans + 1
			else:
				ans = ans + 2
			if a != b and b != n:
				if a == 2:
					ans = ans + 1
				else:
					ans = ans + 2
	print(ans)


import math

def primeFac(n):
	c = 0
	for i in range(1, int(math.sqrt(n)) + 1):
		if n % i == 0:
			if n // i == i:
				c += 1
			else:
				c += 2
	c -= 1
	if n % 2 == 0:
		print(c * 2 - 1)
	else:
		print(c * 2)

def main():
	t = int(input())
	while t:
		n = int(input())
		primeFac(n)
		t -= 1
main()


import math

def solve(n):
	res = 0
	for i in range(1, int(math.sqrt(n)) + 1):
		if n % i == 0:
			(x, y) = (i, n // i)
			if y == 2:
				res += 1
			else:
				res += 2
			if x != y and y != n:
				if x == 2:
					res += 1
				else:
					res += 2
	return res
t = int(input())
for _ in range(t):
	n = int(input())
	print(solve(n))


t = int(input())
for _ in range(0, t):
	n = int(input())
	c = 0
	i = 1
	while i * i <= n:
		if n % i == 0:
			c += 1
			if n - i != i:
				c += 1
			num2 = n // i
			if num2 != n and num2 != i:
				c += 1
				if n - num2 != num2:
					c += 1
		i += 1
	print(c)


from math import sqrt

def factors(n):
	count = 0
	fact = set()
	for i in range(1, int(sqrt(n)) + 2):
		if n % i == 0:
			fact.add(i)
			fact.add(n // i)
			fact.add(n - i)
			fact.add(n - n // i)
	return len(fact)
T = int(input())
for _ in range(T):
	n = int(input())
	print(factors(n) - 2)


from functools import reduce

def factors(n):
	return set(reduce(list.__add__, ([i, n // i] for i in range(1, int(n ** 0.5) + 1) if n % i == 0)))
for j in range(int(input())):
	n = int(input())
	out = (len(factors(n)) - 1) * 2
	if n % 2 == 0:
		out -= 1
	print(out)


for _ in range(int(input())):
	n = int(input())
	count = 0
	for i in range(1, int(pow(n, 0.5)) + 1):
		if n % i == 0:
			(f1, f2) = (i, n // i)
			count += 1 if f2 == 2 else 2
			if f1 != f2 and f2 != n:
				count += 1 if f1 == 2 else 2
	print(count)


for _ in range(int(input())):
	n = int(input())
	i = 1
	count = 0
	while i * i <= n:
		if n % i == 0:
			count += 2
		i += 1
	print(2 * count - 2 - (n % 2 == 0) - 2 * ((i - 1) ** 2 == n))


import math

def printDivisors(n):
	count = 0
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				count += 1
			else:
				count += 2
		i = i + 1
	return count
for _ in range(int(input())):
	n = int(input())
	if n % 2 == 0:
		print(2 * (printDivisors(n) - 1) - 1)
	else:
		print(2 * (printDivisors(n) - 1))


import math
alpha = 'abcdefghijklmnopqrstuvwxyz'
for test in range(int(input())):
	n = int(input())
	i = 1
	divisor = 0
	while i <= math.sqrt(n):
		if i != n:
			if n % i == 0:
				divisor += 1 + (n // i != i)
		i += 1
	count = (divisor - 1) * 2
	if n % 2 == 0:
		print(count - 1)
	else:
		print(count)


from math import *
t = int(input())
for i in range(t):
	n = int(input())
	if n == 2:
		print(1)
		continue
	lst = []
	ctr = 0
	for i in range(1, int(sqrt(n)) + 1):
		if n % i == 0:
			lst.append(i)
			if not sqrt(n) == i:
				lst.append(n // i)
	for j in lst:
		if j <= int(ceil(n / 2)):
			ctr += 1
	ctr = ctr * 2
	if n % 2 == 0:
		ctr -= 1
	print(ctr)


import sys

def List():
	return list(map(int, sys.stdin.readline().split()))

def Ints():
	return map(int, sys.stdin.readline().split())

def Strs():
	return sys.stdin.readline().strip().split()

def Op(s):
	sys.stdout.write(s + '\n')

def Int():
	return int(sys.stdin.readline())

def Str():
	return sys.stdin.readline().strip()

def NList(n, l):
	return [n for i in range(l)]

def gcd(a, b):
	if b == 0:
		return a
	return gcd(b, a % b)

def lcm(a, b):
	return a * b / gcd(a, b)
for _ in range(Int()):
	n = Int()
	c = 0
	for i in range(1, int(n ** (1 / 2)) + 1):
		if n % i == 0:
			if n // i == i:
				c += 1
				continue
			c += 2
	if n % 2 == 0:
		print((c - 1) * 2 - 1)
	else:
		print((c - 1) * 2)


import math
T = int(input())
for t in range(T):
	n = int(input())
	a = []
	i = 1
	while i <= math.sqrt(n):
		if n % i == 0:
			if n / i == i:
				a.append(i)
			else:
				a.append(i)
				a.append(n / i)
		i = i + 1
	c = len(a) - 1
	if n % 2 == 0:
		print(c * 2 - 1)
	else:
		print(c * 2)


import math
from bisect import bisect, bisect_left, bisect_right
import heapq
from math import gcd, floor, sqrt, log
from collections import defaultdict as ddc
from collections import Counter

def intin():
	return int(input())

def mapin():
	return map(int, input().split())

def strin():
	return input().split()
INF = 10 ** 20
mod = 1000000007

def LIS(arr, n):
	dp = [10 ** 9] * (n + 1)
	for ele in arr:
		dp[bisect_left(dp, ele)] = ele
	return bisect_left(dp, 10 ** 9)

def exponentiation(bas, exp, mod=1000000007):
	t = 1
	while exp > 0:
		if exp % 2 != 0:
			t = t * bas % mod
		bas = bas * bas % mod
		exp //= 2
	return t % mod

def MOD(p, q=1, mod=1000000007):
	expo = 0
	expo = mod - 2
	while expo:
		if expo & 1:
			p = p * q % mod
		q = q * q % mod
		expo >>= 1
	return p
yes = 'YES'
no = 'NO'
even = 'EVEN'
odd = 'ODD'

def process(n):
	ans = 0
	for i in range(1, int(sqrt(n)) + 1):
		if not n % i:
			(a, b) = (i, n // i)
			ans += 2 if a != b else 1
	ans -= 1
	if n % 2:
		return 2 * ans
	return 2 * ans - 1

def main():
	for _ in range(int(input())):
		n = intin()
		print(process(n))
main()


import math
for t in range(int(input())):
	n = int(input())
	if n == 2:
		print(1)
	elif n == 3:
		print(2)
	else:
		count = 2
		l = []
		for i in range(2, int(math.sqrt(n)) + 1):
			if n % i == 0:
				if n // i == i:
					l.append(i)
				else:
					l.append(i)
					l.append(n // i)
		for i in l:
			if i == n // 2:
				count += 1
			else:
				count += 2
		print(count)


import math

def factors(n):
	c = 0
	for i in range(1, int(math.sqrt(n)) + 1):
		if n % i == 0:
			c += 1
			if n // i != i:
				c += 1
	return c
for _ in range(int(input())):
	n = int(input())
	ans = 0
	if n == 3:
		print(2)
		continue
	ans = factors(n) - 1
	ans *= 2
	if n % 2 == 0:
		ans -= 1
	print(ans)