Goal: To incorporate the density function effectively into the Sieve of Hotchkiss, to manage not only the prime candidates (A and B sets) but also handle the composites from the product sets (AA, AB, BB).

**Theorem Overview:**

**Prime Candidates**:- A = {6k – 1 | k ∈ Z}
- B = {6k + 1 | k ∈ Z}

**Composite Candidates**:- AA = {(6x – 1)(6y – 1) | x, y ∈ Z}
- AB = {(6x – 1)(6y + 1) | x, y ∈ Z}
- BB = {(6x + 1)(6y + 1) | x, y ∈ Z}

**Enhanced Implementation**

**Density Function**:- The heuristic
`2 / (3 * np.log(n))`

is used to adjust computational efforts based on the prime density. This function helps focus on areas where prime candidates are denser.

- The heuristic
**Segmented Sieving**:- The sieve processes numbers in segments, which optimizes memory usage and cache performance by dividing the range into manageable parts (
`segment_size = int(np.sqrt(n)) + 1`

).

- The sieve processes numbers in segments, which optimizes memory usage and cache performance by dividing the range into manageable parts (

**Here’s the enhanced implementation:**

The Hotchkiss Prime Sieve has been enhanced with several key features to improve its efficiency, including the density function and segmented sieving.

The Python code below incorporates the discussed enhancements, focusing on prime candidates (A and B sets) and efficiently handling the composites from product sets (AA, AB, BB).

import numpy as np

def density_function(n):

return 2 / (3 * np.log(n)) if n > 1 else 0

def optimized_sieve_of_hotchkiss(n):

if n < 2:

return 0, [], [], []

primes_a = set()

primes_b = set()

primes_combined = set()

is_composite = [False] * (n + 1)

# Initialize primes_combined with 2 and 3

if n >= 2:

primes_combined.add((2, 'A'))

primes_a.add(2)

if n >= 3:

primes_combined.add((3, 'B'))

primes_b.add(3)

segment_size = int(np.sqrt(n)) + 1

for low in range(2, n + 1, segment_size):

high = min(low + segment_size - 1, n)

density = density_function(high)

# Estimate the number of primes in the segment using the density function

for k in range(1, int(density * high // 6) + 2):

p1 = 6 * k - 1

p2 = 6 * k + 1

if p1 > high:

break

if p1 >= low and p1 <= high and not is_composite[p1]:

primes_a.add(p1)

primes_combined.add((p1, 'A'))

for multiple in range(p1 * p1, n + 1, p1):

is_composite[multiple] = True

if p2 >= low and p2 <= high and not is_composite[p2]:

primes_b.add(p2)

primes_combined.add((p2, 'B'))

for multiple in range(p2 * p2, n + 1, p2):

is_composite[multiple] = True

# Ensure all possible multiples in the current range are marked

for k in range(1, int(high // 6) + 2):

a1 = 6 * k - 1

b1 = 6 * k + 1

if a1 <= high and a1 >= low:

for multiple in range(a1 * a1, n + 1, a1):

is_composite[multiple] = True

if b1 <= high and b1 >= low:

for multiple in range(b1 * b1, n + 1, b1):

is_composite[multiple] = True

# After marking, collect remaining primes and apply the Hotchkiss Condition

for num in range(5, n + 1):

if not is_composite[num]:

if num % 6 == 5:

# Check the Hotchkiss condition for numbers in set A

if not is_divisible_by_product(num, primes_a, primes_b):

primes_a.add(num)

primes_combined.add((num, 'A'))

elif num % 6 == 1:

# Check the Hotchkiss condition for numbers in set B

if not is_divisible_by_product(num, primes_a, primes_b):

primes_b.add(num)

primes_combined.add((num, 'B'))

# Convert sets to lists and sort them

primes_combined = sorted(primes_combined, key=lambda x: x[0])

primes_a = sorted(primes_a)

primes_b = sorted(primes_b)

total_primes = len(primes_combined)

return total_primes, primes_a, primes_b, primes_combined

def is_divisible_by_product(num, primes_a, primes_b):

"""Checks if a number is divisible by any product of primes in sets A or B."""

for p1 in primes_a:

for p2 in primes_b:

if num % (p1 * p2) == 0:

return True

for p1 in primes_a:

for p2 in primes_a:

if num % (p1 * p2) == 0:

return True

for p1 in primes_b:

for p2 in primes_b:

if num % (p1 * p2) == 0:

return True

return False

# Example usage:

n = 100

total, primes_a, primes_b, primes_combined = optimized_sieve_of_hotchkiss(n)

print(f"Total primes up to {n}: {total}")

print(f"Primes in form A: {primes_a}")

print(f"Primes in form B: {primes_b}")

print(f"All primes: {primes_combined}")