Question

Suppose we select some initial number ? and then build the sequence of values by the following 0

Hint: each time you find a multiple of each picked number in the initial list [1:n], make it zero. Elements

not equal to zero are primes. This works a lot better than trying to ‘erase’ the element. Because each

time you erase an element the remaining indexes roll over, i.e., they change.

1. Find the possible ranges of a and b for which a+b+c is around 300 (include the paper & pencil analysis at the end of your report)

2. Compute c for each combination of a and b

3. Verifyif?+?+?=300 and?2 +?2 =?2 and?<?<?

4. Print results

rules:

? =1? forvaluesof? even ? 2?−1 ?−1

? =1(3? +1)forvaluesof ? odd ? 2 ?−1 ?−1

As long as you reach 1 the sequence must stop. If ? is odd, the next element in the sequence grows, and if it is ?−1

even, the next element in the sequence decreases. With different starting values, examples of the generated sequences are:

111 22,1 2

3 3, 5, 8, 4, 2, 1 8

4 4, 2, 1 4

Starting value

? 0

?? ?? ?? ?? ...

Maximum value in the sequence

Page | 5

5 5, 8, 4, 2, 1 8

6 6, 3, 5, 8, 4, 2, 1 8

7 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1 26

If you decide not to stop, after the sequence reaches 1 the sequence enters the infinite loop as 1 2 1 2 1 2. You need to avoid this effect in your code. You don’t want an infinite loop.

Write code to

1. Compute the elements of the sequence for starting values from 1 to 25. Make a plot of the starting value

versus the number of elements in the sequence to get 1. E.g., if the starting ? = 7, the number of 0

elements in the sequence are 12, these are: 7, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1. An example of the plot is:

2. Calculate what is the maximum value reached in the sequence for each starting value.

Answer 1

# Pythogoreon Triplets
sum = 300
# For all values of a from (0, 300)
for a in range(sum + 1):
    # For all values of b > a and b < 300
    for b in range(a + 1, sum + 1):
        # For all values of c > b and c < 300
        for c in range(b + 1, sum + 1):
            # If sum(a + b + c) == 300 and a^2 + b^2 = c^2
            if (a + b + c == sum) and (a**2 + b**2 == c**2):
                # Print the numbers
                print(f'A : {a}, B : {b}, C : {c}')

# functions to calculate T based on odd or even
def calculateTEven(Ti):
    return int(Ti * 0.5)
def calculateTOdd(Ti):
    return int(0.5 * (3 * Ti + 1))

sequences = []
for T in range(1, 26):
    T0 = T
    seq = []
    while True:
        seq.append(T0)
        if T0 == 1:
            break
        if(T0 % 2 == 0):
            T0 = calculateTEven(T0)
        else:
            T0 = calculateTOdd(T0)
    sequences.append(seq)

print(f'T0\tMaximum\t\tSeries')
for i in range(1, 26):
    print(f'{i}\t   {max(sequences[i - 1])}\t\t{sequences[i - 1]}')

import matplotlib.pyplot as plt
# get the number of elements in each sequence
seq_lengths = [len(seq) for seq in sequences]
# get the T0 (1 to 25)
starting_numbers = [i for i in range(1, 26)]
# plot the graph
plt.plot(starting_numbers, seq_lengths)
# Add title
plt.title("Starting Number VS Number of Elements in Seq")
# Add X label
plt.xlabel("Starting Number")
# Add Y Label
plt.ylabel("Number of Elements per sequence")
# Display the plot
plt.show()