# This tutorial shows how to create a function in Python in one .py file and
# then import it as a module and execute it in another .py Python script.

# The first example function will be used to calculate the factorial of n.
# We will call this function 'fact' and it will be saved in the script
# mod1.py'.

# The format for entering this function is shown in the lines below.
# They are commented out in this script, but these lines of code appear in
# the file mod1.m.

# def fact(n):
#     z = 1
#     for i in range(n):
#         z = z*(i+1)
#     return z

# To call an execute this function, the file mod1.py must me in the same
# directory as the current script that you're working in.  Then, simply
# import mod1.py and then call the function using `mod1.fact(n)' where n is 
# the integer that you want to take the factorial of.
import mod1

# Here's the factorial of 5.
x = mod1.fact(5)
x

120

# Here's the factorial of 6.
x = mod1.fact(6)
x

720

# mod1.py also has a function to calculate the number of combinations for
# selecting m objects for n.  c = n!/((n - m)!m!).  Here's the code that is
# contained in mod1.py.  Note that the 'choose' function relies on the 'fact' 
# function that is also contained in mod1.py.

# def choose(n, m):
#     c = fact(n)/(fact(n - m)*fact(m))
#     return int(c)

# Here's 6 choose 3.
x = mod1.choose(6, 3)
x

20

# Of course, a factorial function already exists in the 'math' module and we didn't need to write 
# our own.
import math
math.factorial(5), math.factorial(6)

(120, 720)

# There is also a module/function for calculating combinations.
from scipy.special import comb
comb(6, 3, exact = True)

20

# Here's one more basic example.  We will calculate the quadrature sum of an
# array of numbers.  The array will be passed to the function which will return
# the scaler quadrature sum.  This type of calculation is used often in 
# propagation of errors.  It is also used to find the magnitude of a vector.

# The following code is included in mod1.py:
# def quad(xx):
#     total = 0
#     for i in range(len(xx)):
#         total = total + xx[i]**2
#     return total**0.5

# Here's the quadrature sum of 1, 2, 3: sqrt(1^2 + 2^2 + 3^2) = sqrt(14)
xx = [1, 2, 3]
mag = mod1.quad(xx)
mag

3.7416573867739413

# As a final example, we'll create a slightly more complicated function.
# This function, called 'primes(a, b)' in the module mod2.py, will find all 
# of the prime numbers between input integers a and b.  The function code 
# is shown below without any explanations.  If you open the file mod2.py you 
# will find some comments that help explain the purpose of the
# various lines of code.  The function returns to values.  The first is an
# array of the prime numbers found is sequential order.  The second item returned
# is the number of primes that were found.  The code also checks that the user
# supplied appropriate inputs.  The requirements are that a and b are positive
# integers and that b > a.

#def primes(a, b):
#     import numpy as np
#     primes = []
#     if a % 1 == 0 and b % 1 == 0 and a > 0 and b > 0 and b > a:
#         x = 0
#         for i in range (a, b + 1):
#             cnt = 0
#             if i != 1 and i % 2 == 1 and sum(map(int, str(i))) % 3 != 0\
#             and i % 10 != 5 or i == 2 or i == 3 or i == 5:
#                 j = 7
#                 while j < i/(j - 2) and cnt == 0:
#                     if i % j == 0:
#                         cnt = 1
#                     j = j + 2
#                 if cnt == 0:
#                     x = x + 1
#                     primes = primes + [i]
#         f = np.array(primes)
#         n = len(f)
#     else:
#         f = 'ERROR: a and b in primes(a, b) must be positive integers with b > a.'
#         n = -1
#     return f, n 

# Here's what happens when mod2.primes(a, b) is called with bad inputs.
import mod2
xx, x = mod2.primes(6.2, -4)
print(xx)
print(x)

ERROR: a and b in primes(a, b) must be positive integers with b > a.
-1

# Here's a couple of simple tests with sensible inputs.
a = 1
b = 100
xx, x = mod2.primes(a, b)
print('There are', x, 'prime numbers between', a, 'and', b)
xx

There are 25 prime numbers between 1 and 100

array([ 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])

a = 500
b = int(10e3)
xx, x = mod2.primes(a, b)
print('There are', x, 'prime numbers between', a, 'and', b)
np.set_printoptions(threshold = 2000) # Tell Python to show the full array.
xx

There are 1134 prime numbers between 500 and 10000

array([ 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])

# Let's try calling the function with some more interesting inputs.
a = 92939223
b = 92939600
xx, x = mod2.primes(a, b)
print('There are', x, 'prime numbers between', a, 'and', b)
xx

There are 16 prime numbers between 92939223 and 92939600

array([92939243, 92939261, 92939299, 92939311, 92939321, 92939377,
       92939387, 92939401, 92939453, 92939467, 92939479, 92939501,
       92939533, 92939543, 92939563, 92939593])

# We can use our prime number function to find the number of prime numbers
# between, say, 1 and 1e6 and write those numbers to a file.
from datetime import datetime
max_n = int(1e6)
print(datetime.now())
xx, x = mod2.primes(1, max_n)
print(datetime.now())
print(x)
import numpy as np
np.savetxt("Jupyter primes_1e6.txt", xx, fmt="%s")
# This took about 15 seconds.

2021-03-11 09:33:18.194563
2021-03-11 09:33:33.894790
78498

# Here's a plot of the number of prime numbers less than x versus x.
import matplotlib.pyplot as plt
plt.plot(xx, np.arange(1, len(xx) + 1, 1), linewidth = 2)
plt.xlabel('x')
plt.ylabel('Number of primes less than x')
plt.axis((0, 1e6, 0, 8e4));