Sieve of Sundaram

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.^[1]^[2]

Algorithm

The sieve starts with a list of the integers from 1 to $n$ . From this list, all numbers of the form $i + j + 2 ij$ are removed, where $i$ and $j$ are positive integers such that $1 \leq i \leq j$ and $i + j + 2 ij \leq n .$ The remaining numbers are doubled and incremented by one, giving a list of the odd prime numbers (that is, all primes except 2) below $2 n + 2.$

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 $2 i + 1$ , Sundaram's method crosses out $i + j (2 i + 1)$ for $1 \leq j \leq ⌊ k /2 ⌋ .$

Correctness

If we start with integers from $1$ to $n$ , the final list contains only odd integers from $3$ to $2 n + 1.$ From this final list, some odd integers have been excluded; we must show these are precisely the composite odd integers less than $2 n + 2$ .

Let $q$ be an odd integer of the form $2 k + 1.$ Then, $q$ is excluded if and only if $k$ is of the form $i + j + 2 ij$ , that is $q = 2(i + j + 2 ij) + 1.$ Then $q = (2 i + 1)(2 j + 1).$

So, an odd integer is excluded from the final list if and only if it has a factorization of the form $(2 i + 1)(2 j + 1)$ — 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 $2 n + 2.$

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 <= k:
            integers_list[i + j + 2 * i * j] = False
            j += 1
    if n > 2:
        print(2, end=' ')
    for i in range(1, k + 1):
        if integers_list[i]:
            print(2 * i + 1, end=' ')

Asymptotic complexity

The above obscure-but-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 cannot perform any composite number culling; the proper range is to the array indices that represent odd numbers less than $\sqrt n$ .
The code does not properly account for indexing of Python arrays, which are zero-based so that it ignores the values at the bottom and top of the array; this is a minor issue, but serves to show that the algorithm behind the code has not been clearly understood.
The inner culling loop (the j loop) exactly reflects the way the algorithm is formulated, but seemingly without realizing that the indexed culling starts at exactly the index representing the square of the base odd number and that the indexing using multiplication can much more easily be expressed as a simple repeated addition of the base odd number across the range; in fact, this method of adding a constant value across the culling range is exactly how the Sieve of Eratosthenes culling is generally implemented.

The following Python code in the same style resolves the above three issues, as well converting the code to a prime-counting function that also displays the total number of composite-culling operations:

from math import isqrt
 
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."""
    if n < 3:
        if n < 2:
            return 0
        else:
            return 1    
    k = (n - 3) // 2 + 1

    integers_list = [True for i in range(k)]

    ops = 0

    for i in range((isqrt(n) - 3) // 2 + 1):
#        if integers_list[i]: # adding this condition turns it into a SoE!
            p = 2 * i + 3
            s = (p * p - 3) // 2 # compute cull start

            for j in range(s, k, p):
                integers_list[j] = False
                ops += 1

    print("Total operations:  ", ops, ";", sep='')

    count = 1
    for i in range(k):
        if integers_list[i]:
            count += 1

    print("Found ", count, " primes to ", n, ".", sep='')

The commented-out line is all that is necessary to convert the Sieve of Sundaram to the Odds-Only Sieve of Eratosthenes; this clarifies that the only difference between these two algorithms is that the Sieve of Sundaram culls composite numbers using all odd numbers as the base values, whereas the Odds-Only Sieve of Eratosthenes uses only the odd primes as base values, with both ranges of base values bounded to the square root of the range.

When run for various ranges, it is immediately clear that while, of course, the resulting count of primes for a given range is identical between the two algorithms, the number of culling operations is much higher for the Sieve of Sundaram and also grows much more quickly with increasing range.

From the above implementation, it is clear that the amount of work done is given by

$\int _{a}^{b}{\frac {n}{2x}}\,dx,$

where $n$ is the range to be sieved and the interval $[a, b]$ is the odd numbers between 3 and $\sqrt n$ . (The interval $[a, b]$ actually starts at the square of the odd base values, but this difference is negligible for large ranges.)

As the integral of the reciprocal of $x$ is exactly $log(x)$ , and as the lower value for $a$ is relatively very small (close to 1, whose logarithm is 0), this is about

${\frac {n}{8}}\log(n).$

Ignoring the constant factor of 1/8, the asymptotic complexity is clearly $O (n log(n)).$

References

^ V. Ramaswami Aiyar (1934). "Sundaram's Sieve for Prime Numbers". The Mathematics Student. 2 (2): 73. ISSN 0025-5742.
^ G. (1941). "Curiosa 81. A New Sieve for Prime Numbers". Scripta Mathematica. 8 (3): 164.

External links

A C99 implementation of the Sieve of Sundaram using bitarrays

[1] V. Ramaswami Aiyar (1934). "Sundaram's Sieve for Prime Numbers". The Mathematics Student. 2 (2): 73. ISSN 0025-5742.

[2] G. (1941). "Curiosa 81. A New Sieve for Prime Numbers". Scripta Mathematica. 8 (3): 164.

[1]

[2]

v t e Number-theoretic algorithms
Primality tests	AKS APR Baillie–PSW Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius Solovay–Strassen Miller–Rabin
Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Pritchard Sieve of Sundaram Wheel factorization
Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks's square forms Trial division Shor's
Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's
Euclidean division	Binary Chunking Fourier Goldschmidt Newton-Raphson Long Short SRT
Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve
Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's
Modular square root	Cipolla Pocklington's Tonelli–Shanks Berlekamp Kunerth
Other algorithms	Chakravala Cornacchia Exponentiation by squaring Integer square root Integer relation (LLL; KZ) Modular exponentiation Montgomery reduction Schoof Trachtenberg system
Italics indicate that algorithm is for numbers of special forms