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package factors
import "fmt"
type DigitType struct {
regularity uint8
totativeType TotativeType
}
type TotativeType byte
const (
// This number does not have any totative factors
Regular TotativeType = 0xC0
// This number's totative part is divisible by (r - 1)
// - this gives it the simplest possible decimal expansion
// for a non-regular (1 digit repeating) and a simple divisibility
// test (sum digits, like 3 or 9 in decimal)
Omega TotativeType = 0xA0
// This number's totative part is divisible by (r + 1)
// - this makes it slightly more complicated than omega
Alpha TotativeType = 0x80
// This number's totative part is divisible by (r^2 - 1)
// but not (r + 1) or (r - 1)
// - these totatives straddle the line between simple and complex
Pseudoneighbour TotativeType = 0x60
// This number's totative part is not divisible by (r^2 - 1)
// - it will not be nice to work with
Opaque TotativeType = 0x40
// This number is zero, and doesn't have a true totative type.
// (except in radix zero, where zero is considered a factor)
Zero TotativeType = 0x00
)
// Zero and one will always have these types.
var (
zeroType = DigitType{regularity: 0, totativeType: Zero}
oneType = DigitType{regularity: 0, totativeType: Regular}
)
// Above this number, regularity indices will be shown as '+'
// instead of the number; this is to keep the code 2 characters long
const maxDisplayRegularity = 7
// The regularity index of a number in a radix, the smallest power of the
// radix that is divisible by the number's regular part.
func (dt DigitType) Regularity() uint8 { return dt.regularity }
// The type of a number's totative part in a radix
func (dt DigitType) TotativeType() TotativeType { return dt.totativeType }
// String returns a string representation of the digit type
// as a two-character code - the first representing regularity,
// the second totative type
func (dt DigitType) String() string {
var rString string
if dt.regularity > maxDisplayRegularity {
rString = "+"
} else {
rString = fmt.Sprint(dt.regularity)
}
var tString string
switch dt.totativeType {
case Zero:
tString = "0"
case Regular:
tString = "R"
case Omega:
tString = "ω"
case Alpha:
tString = "α"
case Pseudoneighbour:
tString = "N"
case Opaque:
tString = "P"
}
return rString + tString
}
// Splits a digit in a radix into its regular and totative parts
func splitPF(digit, radix PrimeFactorization) (regular, totative uint) {
regular, totative = 1, 1
for p, e := range digit.exponents {
if radix.exponents[p] != 0 {
regular *= uintpow(p, e)
} else {
totative *= uintpow(p, e)
}
}
return regular, totative
}
// Splits a digit in a radix into its regular and totative parts
func Split(digit, radix uint) (regular, totative uint) {
return splitPF(PrimeFactorize(digit), PrimeFactorize(radix))
}
// Calculates the regularity index of a number's regular part
func calcRegularity(regular, radix uint) uint8 {
if regular == 0 && radix == 0 {
return 1
}
regularity, radixPower := uint8(0), uint(1)
for radixPower%regular != 0 {
regularity++
radixPower *= radix
}
return regularity
}
// Calculates the totative type of a number's totative part
func calcTotativeType(totative, radix uint) TotativeType {
switch true {
case totative == 0:
return Zero
case totative == 1:
return Regular
case (radix-1)%totative == 0:
return Omega
case (radix+1)%totative == 0:
return Alpha
case (radix*radix-1)%totative == 0:
return Pseudoneighbour
default:
return Opaque
}
}
// Gets the regularity and totative type of one digit in a radix
// In radix zero, zero is type 1R, one is type 0R, and everything else is 0P.
func GetDigitType(digit, radix uint) DigitType {
radixPF := PrimeFactorize(radix)
digitPF := PrimeFactorize(digit)
regular, totative := splitPF(digitPF, radixPF)
regularity := calcRegularity(regular, radix)
totativeType := calcTotativeType(totative, radix)
return DigitType{regularity, totativeType}
}
// Gets the regularity and totative type of each digit in a radix
func DigitMap(radix uint) []DigitType {
radixPF := PrimeFactorize(radix)
types := make([]DigitType, radix, radix)
if radix > 0 {
types[0] = zeroType
}
if radix > 1 {
types[1] = oneType
}
for d := uint(2); d < radix; d++ {
digitPF := PrimeFactorize(d)
regular, totative := splitPF(digitPF, radixPF)
regularity := calcRegularity(regular, radix)
totativeType := calcTotativeType(totative, radix)
types[d] = DigitType{regularity, totativeType}
}
return types
}
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