mathtools/internal/lib/divisor.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

import (
	"math/big"
)

/*
DivisorSummatory computes the sum of sigma(k) for k=1 to n, where sigma is the divisor sum function.
*/
func DivisorSummatory(n *big.Int) *big.Int {
	// TODO: expand to work for different sigmas

	// employing Dirichlet's hyperbola method
	sqrt := new(big.Int).Sqrt(n)

	total := big.NewInt(0)
	for x := big.NewInt(1); x.Cmp(sqrt) <= 0; x.Add(x, big.NewInt(1)) {
		z := new(big.Int).Div(n, x)
		total.Add(total, z)
	}

	total.Mul(total, big.NewInt(2))

	sqrt.Exp(sqrt, big.NewInt(2), nil)
	total.Sub(total, sqrt)

	return total
}