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Concept
Sieve of Sundaram
Summary
In mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up to a specified integer. It was discovered by Indian student S. P. Sundaram in 1934.
Start with a list of the integers from 1 to n. From this list, remove all numbers of the form i + j + 2ij where:
The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (i.e., all primes except 2) below 2n + 2.
The sieve of Sundaram sieves out the composite numbers just as the sieve of Eratosthenes does, but even numbers are not considered; the work of "crossing out" the multiples of 2 is done by the final double-and-increment step. Whenever Eratosthenes' method would cross out k different multiples of a prime 2i+1, Sundaram's method crosses out i + j(2i+1) for .
If we start with integers from 1 to n, the final list contains only odd integers from 3 to 2n + 1. From this final list, some odd integers have been excluded; we must show these are precisely the composite odd integers less than 2n + 2.
Let q be an odd integer of the form 2k + 1. Then, q is excluded if and only if k is of the form i + j + 2ij, that is q = 2(i + j + 2ij) + 1. Then we have:
So, an odd integer is excluded from the final list if and only if it has a factorization of the form (2i + 1)(2j + 1) — which is to say, if it has a non-trivial odd factor. Therefore the list must be composed of exactly the set of odd prime numbers less than or equal to 2n + 2.
def sieve_of_Sundaram(n):
"""The sieve of Sundaram is a simple deterministic algorithm for finding all the prime numbers up to a specified integer."""
k = (n - 2) // 2
integers_list = [True] * (k + 1)
for i in range(1, k + 1):
j = i
while i + j + 2 * i * j 2:
print(2, end=' ')
for i in range(1, k + 1):
if integers_list[i]:
print(2 * i + 1, end=' ')
The above obscure but as commonly implemented Python version of the Sieve of Sundaram hides the true complexity of the algorithm due to the following reasons:
The range for the outer i looping variable is much too large, resulting in redundant looping that can't perform any composite number representation culling; the proper range is to the array index represent odd numbers less than the square root of the range.
Official source
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