Optimization for finding perfect-square algorithm

Question

The question I\'m working on is:

Find which sum of squared factors are a perfect square given a specific range. So if the range was (1..10) you woul

Answer 1

The trick that frequently solves questions like this is to switch from trial division to a sieve. In Python (sorry):

def list_squared(m, n):
    factor_squared_sum = {i: 0 for i in range(m, n + 1)}
    for factor in range(1, n + 1):
        i = n - n % factor  # greatest multiple of factor less than or equal to n
        while i >= m:
            factor_squared_sum[i] += factor ** 2
            i -= factor
    return {i for (i, fss) in factor_squared_sum.items() if isqrt(fss) ** 2 == fss}


def isqrt(n):
    # from http://stackoverflow.com/a/15391420
    x = n
    y = (x + 1) // 2
    while y < x:
        x = y
        y = (x + n // x) // 2
    return x

The next optimization is to step factor only to isqrt(n), adding the factor squares in pairs (e.g., 2 and i // 2).