Step by Step solution about How to resolve the algorithm The sieve of Sundaram step by step in the Mathematica/Wolfram Language programming language

Code Overview

The provided Wolfram code defines a function called SieveOfSundaram that implements a prime number generation algorithm known as the Sieve of Sundaram. It calculates prime numbers up to a specified limit n.

Algorithm Explanation

The Sieve of Sundaram is an efficient algorithm for generating prime numbers. It starts with a range of integers from 1 to n and then iteratively eliminates composite (non-prime) numbers. The algorithm works as follows:

1. Initialize range: Start with an array of True values for numbers from 1 to n/2. This represents the potential prime numbers.

2. Sieve process: For each number i from 1 to n/2:

   • Calculate prefac as 2i + 1.
   • If i + i * prefac is within the range, mark the corresponding number as False in the array. This eliminates composite numbers that can be formed from i and prefac.

3. Generate primes: Gather the positions of True values in the array. Multiply each position by 2 and add 1 to obtain the list of prime numbers.

Code Details

Function Definition:

ClearAll[SieveOfSundaram]
SieveOfSundaram[n_Integer] := Module[{i, prefac, k, ints},

• ClearAll[SieveOfSundaram]: Clears the definition of the SieveOfSundaram function to ensure a fresh start.
• SieveOfSundaram[n_Integer] := Module[{...}]: Defines the SieveOfSundaram function that takes an integer n as a parameter.

Sieving Process:

k = Floor[(n - 2)/2];
ints = ConstantArray[True, k + 1];
Do[
  prefac = 2 i + 1;
  If[i + i prefac <= k,
   ints[[i + i prefac ;; ;; prefac]] = False
   ];
  ,
  {i, 1, k + 1}
  ];

• k = Floor[(n - 2)/2]: Calculates half of the range of integers to sieve.
• ints = ConstantArray[True, k + 1]: Initializes the array of potential prime numbers.
• Do[..., {i, 1, k + 1}]: Iterates through each number i in the range.
• prefac = 2 i + 1: Calculates the prefac value.
• If[i + i prefac <= k, ... ]: Checks if i and prefac are within the range. If so, it eliminates composite numbers formed from them.

Prime Generation:

2 Flatten[Position[ints, True]] + 1

• Position[ints, True]: Finds the positions of True values in the ints array, representing potential prime numbers.
• Flatten[...]: Combines the lists of positions into a single list.
• 2 [...] + 1: Multiplies each position by 2 and adds 1 to generate the list of prime numbers.

Usage Examples:

SieveOfSundaram[600][[;; 100]]
SieveOfSundaram[16000000][[10^6]]

These examples demonstrate the usage of the SieveOfSundaram function. The first call generates and prints the first 100 prime numbers up to 600, while the second call prints the 1,000,000th prime number up to the limit of 16,000,000.

ClearAll[SieveOfSundaram]
SieveOfSundaram[n_Integer] := Module[{i, prefac, k, ints},
  k = Floor[(n - 2)/2];
  ints = ConstantArray[True, k + 1];
  Do[
   prefac = 2 i + 1;
   If[i + i prefac <= k,
    ints[[i + i prefac ;; ;; prefac]] = False
    ];
   ,
   {i, 1, k + 1}
   ];
  2 Flatten[Position[ints, True]] + 1
  ]
SieveOfSundaram[600][[;; 100]]
SieveOfSundaram[16000000][[10^6]]