add convolve command

README.md

@@ -0,0 +1,36 @@
+# mathtools
+mathtools is a program for computing various mathematical results that would be
+tedious to compute by hand.
+
+## Why?
+Obviously, libraries, software packages, and websites exist for these sort of
+calculations, but there are tradeoffs with any approach. Rather than needing to
+write a script, use a REPL, or load a webpage, this allows for an approach more
+like standard CLI utilities such as [GNU
+factor](https://www.gnu.org/software/coreutils/factor).
+
+Generally, I've opted to implement routines for problems that are best solved
+with *algorithms*, rather than *formulas*. For instance, the quadratic formula,
+while useful, is basically plug and chug, and thus isn't implemented here. On
+the other hand, determining whether a number is prime is a little more tedious
+to do by hand, so it's provided as a routine.
+
+## Available routines
+Available routines include:
+* convergents of a periodic continued fraction
+* solving systems of linear congruences with the Chinese remainder theorem
+* discrete logarithm
+* greatest common divisor
+* primality testing
+* Jacobi symbol
+* modular inverse
+* modular square root
+* multiplicative order
+* integer partitions
+* solving Pell equations
+* primitive root modulo n
+* area of a simple polygon from vertex coordinates
+* sieves for totient function, divisor function, Mobius function, and more
+* repetend of the continued fraction of a square root
+* Stirling numbers
+* summatory functions

cmd/common.go

@@ -0,0 +1,32 @@
+/*
+Copyright © 2026 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 cmd
+
+import (
+	"bufio"
+	"os"
+)
+
+func splitLines(file *os.File) []string {
+	slice := make([]string, 0)
+	scanner := bufio.NewScanner(file)
+	for scanner.Scan() {
+		slice = append(slice, scanner.Text())
+	}
+
+	return slice
+}

cmd/convolve.go

@@ -0,0 +1,110 @@
+/*
+Copyright © 2026 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 cmd
+
+import (
+	"bufio"
+	"fmt"
+	"os"
+	"strconv"
+
+	"github.com/spf13/cobra"
+	"scm.dairydemon.net/filifa/mathtools/internal/lib"
+)
+
+var convolveA string
+var convolveB string
+
+func readSequenceFromFile(filepath string) ([]complex128, error) {
+	// initial length 1 so index math works
+	seq := make([]complex128, 1)
+	file, err := os.Open(filepath)
+	if err != nil {
+		return seq, err
+	}
+	defer file.Close()
+
+	lines := splitLines(file)
+	for _, line := range lines {
+		x, err := strconv.ParseComplex(line, 128)
+		if err != nil {
+			return seq, err
+		}
+
+		seq = append(seq, x)
+	}
+
+	return seq, nil
+}
+
+func convolve(cmd *cobra.Command, args []string) {
+	a, err := readSequenceFromFile(convolveA)
+	if err != nil {
+		cobra.CheckErr(err)
+	}
+
+	b, err := readSequenceFromFile(convolveB)
+	if err != nil {
+		cobra.CheckErr(err)
+	}
+
+	bufStdout := bufio.NewWriter(os.Stdout)
+	defer bufStdout.Flush()
+
+	for i, n := range lib.DirichletConvolve(a, b) {
+		if i == 0 {
+			continue
+		}
+
+		if imag(n) == 0 {
+			fmt.Fprintln(bufStdout, real(n))
+		} else {
+			fmt.Fprintln(bufStdout, n)
+		}
+	}
+}
+
+// convolveCmd represents the convolve command
+var convolveCmd = &cobra.Command{
+	Use:   "convolve",
+	Short: "Compute the Dirichlet convolution of two sequences",
+	Long: `Compute the Dirichlet convolution of two sequences.
+
+Each sequence is provided as a file, where line k of the file gives the kth term of the sequence.
+`,
+	Run: convolve,
+}
+
+func init() {
+	rootCmd.AddCommand(convolveCmd)
+
+	// Here you will define your flags and configuration settings.
+
+	// Cobra supports Persistent Flags which will work for this command
+	// and all subcommands, e.g.:
+	// convolveCmd.PersistentFlags().String("foo", "", "A help for foo")
+
+	// Cobra supports local flags which will only run when this command
+	// is called directly, e.g.:
+	// convolveCmd.Flags().BoolP("toggle", "t", false, "Help message for toggle")
+
+	convolveCmd.Flags().StringVarP(&convolveA, "first", "a", "", "first sequence")
+	convolveCmd.MarkFlagRequired("first")
+
+	convolveCmd.Flags().StringVarP(&convolveB, "second", "b", "", "second sequence")
+	convolveCmd.MarkFlagRequired("second")
+}

cmd/divisors.go

@@ -32,7 +32,7 @@ func divisors(cmd *cobra.Command, args []string) {
 	bufStdout := bufio.NewWriter(os.Stdout)
 	defer bufStdout.Flush()
 
-	ch := sieve.DivisorsSieve(divisorsN, divisorsE, 1000)
+	ch := sieve.Divisors(divisorsN, divisorsE, 1000)
 	for i := 0; ; i++ {
 		v, ok := <-ch
 		if !ok {

cmd/mobius.go

@@ -31,7 +31,7 @@ func mobius(cmd *cobra.Command, args []string) {
 	bufStdout := bufio.NewWriter(os.Stdout)
 	defer bufStdout.Flush()
 
-	ch := sieve.MobiusSieve(mobiusN, 1000)
+	ch := sieve.Mobius(mobiusN, 1000)
 	for i := 0; ; i++ {
 		v, ok := <-ch
 		if !ok {

cmd/multiplicativeOrder.go

@@ -0,0 +1,75 @@
+/*
+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 cmd
+
+import (
+	"fmt"
+	"math/big"
+
+	"github.com/spf13/cobra"
+	"scm.dairydemon.net/filifa/mathtools/internal/lib"
+)
+
+var multiplicativeOrderBase string
+var multiplicativeOrderModulus string
+
+func multiplicativeOrder(cmd *cobra.Command, args []string) {
+	g, ok := new(big.Int).SetString(multiplicativeOrderBase, 10)
+	if !ok {
+		cobra.CheckErr("invalid base " + multiplicativeOrderBase)
+	}
+
+	m, ok := new(big.Int).SetString(multiplicativeOrderModulus, 10)
+	if !ok {
+		cobra.CheckErr("invalid modulus " + multiplicativeOrderModulus)
+	}
+
+	gcd := new(big.Int).GCD(nil, nil, g, m)
+	if gcd.Cmp(big.NewInt(1)) != 0 {
+		cobra.CheckErr("base " + multiplicativeOrderBase + " and modulus " + multiplicativeOrderModulus + " are not coprime")
+	}
+
+	k := lib.MultiplicativeOrder(g, m)
+	fmt.Println(k)
+}
+
+// multiplicativeOrderCmd represents the multiplicativeOrder command
+var multiplicativeOrderCmd = &cobra.Command{
+	Use:   "multiplicative-order",
+	Short: "Compute multiplicative order",
+	Long:  `Compute the multiplicative order of a number given a modulus.`,
+	Run:   multiplicativeOrder,
+}
+
+func init() {
+	rootCmd.AddCommand(multiplicativeOrderCmd)
+
+	// Here you will define your flags and configuration settings.
+
+	// Cobra supports Persistent Flags which will work for this command
+	// and all subcommands, e.g.:
+	// multiplicativeOrderCmd.PersistentFlags().String("foo", "", "A help for foo")
+
+	// Cobra supports local flags which will only run when this command
+	// is called directly, e.g.:
+	// multiplicativeOrderCmd.Flags().BoolP("toggle", "t", false, "Help message for toggle")
+	multiplicativeOrderCmd.Flags().StringVarP(&multiplicativeOrderBase, "base", "g", "", "base")
+	multiplicativeOrderCmd.Flags().StringVarP(&multiplicativeOrderModulus, "modulus", "m", "", "modulus")
+
+	multiplicativeOrderCmd.MarkFlagRequired("base")
+	multiplicativeOrderCmd.MarkFlagRequired("modulus")
+}

cmd/primeOmega.go

@@ -32,7 +32,7 @@ func primeOmega(cmd *cobra.Command, args []string) {
 	bufStdout := bufio.NewWriter(os.Stdout)
 	defer bufStdout.Flush()
 
-	ch := sieve.PrimeOmegaSieve(primeOmegaN, primeOmegaMul, 1000)
+	ch := sieve.PrimeOmega(primeOmegaN, primeOmegaMul, 1000)
 	for i := 0; ; i++ {
 		v, ok := <-ch
 		if !ok {

cmd/quadraticResidues.go

@@ -0,0 +1,75 @@
+/*
+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 cmd
+
+import (
+	"bufio"
+	"fmt"
+	"os"
+
+	"github.com/spf13/cobra"
+	"scm.dairydemon.net/filifa/mathtools/internal/lib/sieve"
+)
+
+var quadraticResiduesN uint
+var quadraticResiduesCoprime bool
+
+func quadraticResidues(cmd *cobra.Command, args []string) {
+	bufStdout := bufio.NewWriter(os.Stdout)
+	defer bufStdout.Flush()
+
+	ch := sieve.QuadraticResidues(quadraticResiduesN, quadraticResiduesCoprime, 1000)
+	for i := 0; ; i++ {
+		v, ok := <-ch
+		if !ok {
+			break
+		}
+
+		if i == 0 {
+			continue
+		}
+
+		fmt.Fprintln(bufStdout, v)
+	}
+}
+
+// quadraticResiduesCmd represents the quadraticResidues command
+var quadraticResiduesCmd = &cobra.Command{
+	Use:   "quadratic-residues",
+	Short: "Compute the number of quadratic residues for all moduli less than n",
+	Long:  `Compute the number of quadratic residues for all moduli less than n.`,
+	Run:   quadraticResidues,
+}
+
+func init() {
+	sieveCmd.AddCommand(quadraticResiduesCmd)
+
+	// Here you will define your flags and configuration settings.
+
+	// Cobra supports Persistent Flags which will work for this command
+	// and all subcommands, e.g.:
+	// quadraticResiduesCmd.PersistentFlags().String("foo", "", "A help for foo")
+
+	// Cobra supports local flags which will only run when this command
+	// is called directly, e.g.:
+	// quadraticResiduesCmd.Flags().BoolP("toggle", "t", false, "Help message for toggle")
+
+	quadraticResiduesCmd.Flags().UintVarP(&quadraticResiduesN, "limit", "n", 0, "upper limit")
+	quadraticResiduesCmd.MarkFlagRequired("limit")
+
+	quadraticResiduesCmd.Flags().BoolVarP(&quadraticResiduesCoprime, "coprime-only", "c", false, "only count residues coprime to the modulus")
+}

cmd/radical.go

@@ -31,7 +31,7 @@ func radical(cmd *cobra.Command, args []string) {
 	bufStdout := bufio.NewWriter(os.Stdout)
 	defer bufStdout.Flush()
 
-	ch := sieve.RadicalSieve(radicalN, 1000)
+	ch := sieve.Radical(radicalN, 1000)
 	for i := 0; ; i++ {
 		v, ok := <-ch
 		if !ok {

cmd/shoelace.go

@@ -17,7 +17,6 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 package cmd
 
 import (
-	"bufio"
 	"fmt"
 	"os"
 	"strconv"
@@ -29,16 +28,6 @@ import (
 
 var shoelaceFile string
 
-func splitLines(file *os.File) []string {
-	slice := make([]string, 0)
-	scanner := bufio.NewScanner(file)
-	for scanner.Scan() {
-		slice = append(slice, scanner.Text())
-	}
-
-	return slice
-}
-
 func readFromFile(filepath string) ([]lib.Point, error) {
 	points := make([]lib.Point, 0)
 	file, err := os.Open(filepath)

cmd/stirling.go

@@ -22,7 +22,7 @@ import (
 	"strconv"
 
 	"github.com/spf13/cobra"
-	"scm.dairydemon.net/filifa/mathtools/internal/lib"
+	"scm.dairydemon.net/filifa/mathtools/lib"
 )
 
 var firstKind bool

cmd/totient.go

@@ -31,7 +31,7 @@ func totient(cmd *cobra.Command, args []string) {
 	bufStdout := bufio.NewWriter(os.Stdout)
 	defer bufStdout.Flush()
 
-	for v := range sieve.TotientSieve(totientN, 1000) {
+	for v := range sieve.Totient(totientN, 1000) {
 		if v == 0 {
 			continue
 		}

go.mod

@@ -2,8 +2,9 @@ module scm.dairydemon.net/filifa/mathtools
 
 go 1.24.4
 
+require github.com/spf13/cobra v1.9.1
+
 require (
 	github.com/inconshreveable/mousetrap v1.1.0 // indirect
-	github.com/spf13/cobra v1.9.1 // indirect
 	github.com/spf13/pflag v1.0.6 // indirect
 )

internal/lib/convolve.go

@@ -0,0 +1,32 @@
+/*
+Copyright © 2026 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 DirichletConvolve(a, b []complex128) []complex128 {
+	c := make([]complex128, min(len(a), len(b)))
+	for i, x := range a {
+		for j, y := range b {
+			if i*j >= len(c) {
+				break
+			}
+
+			c[i*j] += x * y
+		}
+	}
+
+	return c
+}

internal/lib/sieve/common.go

@@ -0,0 +1,36 @@
+/*
+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 sieve
+
+func updateMultiples(sieve []uint, p uint, n uint, additive bool) {
+	for q := p; ; q *= p {
+		// sieve[a*b] = sieve[a] * sieve[b] if gcd(a,b) = 1
+		for i := 2 * q; i < n; i += q {
+			if i%(p*q) != 0 {
+				if additive {
+					sieve[i] += sieve[q]
+				} else {
+					sieve[i] *= sieve[q]
+				}
+			}
+		}
+
+		if p*q >= n {
+			break
+		}
+	}
+}

internal/lib/sieve/divisors.go

@@ -30,29 +30,10 @@ func pow(base uint, exp uint) uint {
 	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
-		}
-		println(q)
-
-		// 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.
+Divisors 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 {
+func Divisors(n uint, x uint, buflen uint) chan uint {
 	sieve := make([]uint, n)
 	sieve[0] = 0
 	for i := uint(1); i < n; i++ {
@@ -69,7 +50,12 @@ func DivisorsSieve(n uint, x uint, buflen uint) chan uint {
 			}
 
 			sieve[i] = pow(i, x) + 1
-			updateMultiples(sieve, x, i, n)
+			for j := i; i*j < n; j *= i {
+				// sigma_x(p^k) = p^(kx) + sigma_x(p^(k-1))
+				sieve[i*j] = pow(i*j, x) + sieve[j]
+			}
+
+			updateMultiples(sieve, i, n, false)
 			ch <- sieve[i]
 		}
 	}()

internal/lib/sieve/mobius.go

@@ -17,9 +17,9 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 package sieve
 
 /*
-MobiusSieve computes mobius(k) for k=1 to n, where mobius is the Mobius function.
+Mobius computes mobius(k) for k=1 to n, where mobius is the Mobius function.
 */
-func MobiusSieve(n uint, buflen uint) chan int {
+func Mobius(n uint, buflen uint) chan int {
 	sieve := make([]int, n)
 	for i := 0; i < int(n); i++ {
 		sieve[i] = i

internal/lib/sieve/primeOmega.go

@@ -16,29 +16,16 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 */
 package sieve
 
-func primeOmegaUpdateMultiples(sieve []uint, p uint, n uint, multiplicity bool) {
-	for q := p; ; q *= p {
-		// omega(a*b) = omega(a) + omega(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
-		}
-
-		if multiplicity {
-			sieve[p*q] = 1 + sieve[q]
-		}
+func updatePowers(sieve []uint, p uint, n uint) {
+	for q := p; p*q < n; q *= p {
+		sieve[p*q] = 1 + sieve[q]
 	}
 }
 
 /*
-PrimeOmegaSieve computes omega(k) for k=1 to n, where omega is the prime omega function. If multiplicity is true, factors are counted with multiplicity.
+PrimeOmega computes omega(k) for k=1 to n, where omega is the prime omega function. If multiplicity is true, factors are counted with multiplicity.
 */
-func PrimeOmegaSieve(n uint, multiplicity bool, buflen uint) chan uint {
+func PrimeOmega(n uint, multiplicity bool, buflen uint) chan uint {
 	sieve := make([]uint, n)
 	for i := uint(0); i < n; i++ {
 		sieve[i] = 0
@@ -54,7 +41,11 @@ func PrimeOmegaSieve(n uint, multiplicity bool, buflen uint) chan uint {
 			}
 
 			sieve[i] = 1
-			primeOmegaUpdateMultiples(sieve, i, n, multiplicity)
+			if multiplicity {
+				updatePowers(sieve, i, n)
+			}
+
+			updateMultiples(sieve, i, n, true)
 			ch <- sieve[i]
 		}
 	}()

internal/lib/sieve/qresidue.go

@@ -0,0 +1,131 @@
+/*
+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 sieve
+
+// NOTE: these formulas come from https://web.archive.org/web/20151224013638/http://www.maa.org/sites/default/files/Walter_D22068._Stangl.pdf
+
+func qrPowersOfTwoCoprime(sieve []uint, n uint) {
+	for q := uint(8); 2*q < n; q *= 2 {
+		// q(2^n) = 2^(n-3) for n >= 3
+		sieve[2*q] = 2 * sieve[q]
+	}
+}
+
+func qrPowersOfOddPrimesCoprime(sieve []uint, p uint, n uint) {
+	for q := p; p*q < n; q *= p {
+		// q(p^n) = (p^n - p^(n-1)) / 2
+		sieve[p*q] = (p*q - q) / 2
+	}
+}
+
+func qrPowersOfTwo(sieve []uint, n uint) {
+	k := 1
+	for q := uint(1); 2*q < n; q *= 2 {
+		if k%2 == 0 {
+			// s(2^n) = (2^(n-1) + 4) / 3 for even n
+			sieve[2*q] = (q + 4) / 3
+		} else {
+			// s(2^n) = (2^(n-1) + 5) / 3 for odd n
+			sieve[2*q] = (q + 5) / 3
+		}
+
+		k += 1
+	}
+}
+
+func qrPowersOfOddPrimes(sieve []uint, p uint, n uint) {
+	k := 3
+	for q := p * p; p*q < n; q *= p {
+		if k%2 == 0 {
+			// s(p^n) = (p^(n+1) + p + 2) / (2*(p+1)) for even n
+			sieve[p*q] = (p*p*q + p + 2) / (2 * (p + 1))
+		} else {
+			// s(p^n) = (p^(n+1) + 2*p + 1) / (2*(p+1)) for odd n
+			sieve[p*q] = (p*p*q + 2*p + 1) / (2 * (p + 1))
+		}
+
+		k += 1
+	}
+}
+
+/*
+QuadraticResidues computes the number of quadratic residues modulo k for k=1 to n.
+
+see https://oeis.org/A000224 and https://oeis.org/A046073
+*/
+func QuadraticResidues(n uint, coprime bool, buflen uint) chan uint {
+	sieve := make([]uint, n)
+	for i := uint(0); i < n; i++ {
+		sieve[i] = 1
+	}
+
+	ch := make(chan uint, buflen)
+	go func() {
+		defer close(ch)
+		if coprime {
+			sieveQRCoprime(sieve, n, ch)
+		} else {
+			sieveQR(sieve, n, ch)
+		}
+	}()
+
+	return ch
+}
+
+func sieveQRCoprime(sieve []uint, n uint, ch chan uint) {
+	for i := uint(0); i < n; i++ {
+		if i == 0 || i == 1 || i == 4 || i == 6 || i == 8 || i == 12 || i == 24 || sieve[i] != 1 {
+			ch <- sieve[i]
+			continue
+		}
+
+		if i == 2 {
+			qrPowersOfTwoCoprime(sieve, n)
+		} else {
+			// q(p) = (p - 1) / 2
+			sieve[i] = (i - 1) / 2
+			qrPowersOfOddPrimesCoprime(sieve, i, n)
+		}
+
+		updateMultiples(sieve, i, n, false)
+		ch <- sieve[i]
+	}
+}
+
+func sieveQR(sieve []uint, n uint, ch chan uint) {
+	for i := uint(0); i < n; i++ {
+		if i == 0 || i == 1 || sieve[i] != 1 {
+			ch <- sieve[i]
+			continue
+		}
+
+		if i == 2 {
+			qrPowersOfTwo(sieve, n)
+		} else {
+			// s(p) = (p + 1) / 2
+			sieve[i] = (i + 1) / 2
+			if i*i < n {
+				// s(p^2) = (p^2 - p + 2) / 2
+				sieve[i*i] = (i*i - i + 2) / 2
+			}
+			qrPowersOfOddPrimes(sieve, i, n)
+		}
+
+		updateMultiples(sieve, i, n, false)
+		ch <- sieve[i]
+	}
+}

internal/lib/sieve/radical.go

@@ -16,28 +16,10 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 */
 package sieve
 
-func radicalUpdateMultiples(sieve []uint, p uint, n uint) {
-	for q := p; ; q *= p {
-		// rad(a*b) = rad(a) * rad(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
-		}
-
-		// rad(p^k) = rad(p)
-		sieve[p*q] = sieve[q]
-	}
-}
-
 /*
-RadicalSieve computes rad(k) for k=1 to n, where rad(n) is the radical of n.
+Radical computes rad(k) for k=1 to n, where rad(n) is the radical of n.
 */
-func RadicalSieve(n uint, buflen uint) chan uint {
+func Radical(n uint, buflen uint) chan uint {
 	sieve := make([]uint, n)
 	sieve[0] = 0
 	for i := uint(1); i < n; i++ {
@@ -54,7 +36,12 @@ func RadicalSieve(n uint, buflen uint) chan uint {
 			}
 
 			sieve[i] = i
-			radicalUpdateMultiples(sieve, i, n)
+			for j := i; i*j < n; j *= i {
+				// rad(p^k) = rad(p)
+				sieve[i*j] = sieve[i]
+			}
+
+			updateMultiples(sieve, i, n, false)
 			ch <- sieve[i]
 		}
 	}()

internal/lib/sieve/totient.go

@@ -17,9 +17,9 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 package sieve
 
 /*
-TotientSieve computes totient(k) for k=1 to n, where totient is Euler's totient function. buflen sets the buffer length of the returned channel. Larger buffer lengths can result in better performance at the cost of higher memory usage.
+Totient computes totient(k) for k=1 to n, where totient is Euler's totient function. buflen sets the buffer length of the returned channel. Larger buffer lengths can result in better performance at the cost of higher memory usage.
 */
-func TotientSieve(n uint, buflen uint) chan uint {
+func Totient(n uint, buflen uint) chan uint {
 	totients := make([]uint, n)
 	totients[0] = 0
 	totients[1] = 1