mathtools/internal/lib/divisors.go

/*
Copyright © 2025 filifa

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
package lib

func pow(base uint, exp uint) uint {
	result := uint(1)
	for exp > 0 {
		if exp%2 == 1 {
			result *= base
		}

		exp >>= 1
		base *= base
	}

	return result
}

func updateMultiples(sieve []uint, x uint, p uint, n uint) {
	for q := p; ; q *= p {
		// sigma_x(a*b) = sigma_x(a) * sigma_x(b) if gcd(a,b) = 1
		for i := 2 * q; i < n; i += q {
			if i%(p*q) != 0 {
				sieve[i] *= sieve[q]
			}
		}

		if p*q >= n {
			break
		}

		// sigma_x(p^k) = p^(kx) + sigma_x(p^(k-1))
		sieve[p*q] = pow(p*q, x) + sieve[q]
	}
}

/*
DivisorSieve computes sigma_x(k) for k=1 to n, where sigma_x is the divisor sum function. x sets the power each divisor is raised to.
*/
func DivisorsSieve(n uint, x uint, buflen uint) chan uint {
	sieve := make([]uint, n)
	sieve[0] = 0
	for i := uint(1); i < n; i++ {
		sieve[i] = 1
	}

	ch := make(chan uint, buflen)
	go func() {
		defer close(ch)
		for i := uint(0); i < n; i++ {
			if i == 0 || i == 1 || sieve[i] != 1 {
				ch <- sieve[i]
				continue
			}

			sieve[i] = pow(i, x) + 1
			updateMultiples(sieve, x, i, n)
			ch <- sieve[i]
		}
	}()

	return ch
}