From a663fa77cde6ad48f29abd6973211c179ff32464 Mon Sep 17 00:00:00 2001
From: Adrien Hopkins <adrien.p.hopkins@gmail.com>
Date: Mon, 4 Sep 2023 18:51:26 -0500
Subject: Increase detail of digit map neighbours

It now distinguishes between omega (factors of r - 1), alpha (factors of
r + 1) and the rest.  The previous categorization determined whether or
not totatives and semitotatives have simple patterns; this new
categorization determines which simple patterns they follow)
---
 factors/digit_map.go | 25 +++++++++++++++++++++----
 1 file changed, 21 insertions(+), 4 deletions(-)

(limited to 'factors/digit_map.go')

diff --git a/factors/digit_map.go b/factors/digit_map.go
index e78cce0..db562db 100644
--- a/factors/digit_map.go
+++ b/factors/digit_map.go
@@ -12,9 +12,18 @@ type TotativeType uint8
 const (
 	// This number does not have any totative factors
 	Regular TotativeType = iota
+	// 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
+	// This number's totative part is divisible by (r + 1)
+	// - this makes it slightly more complicated than omega
+	Alpha
 	// This number's totative part is divisible by (r^2 - 1)
-	// - this makes it easier to work with
-	Neighbour
+	// but not (r + 1) or (r - 1)
+	// - these totatives straddle the line between simple and complex
+	Pseudoneighbour
 	// This number's totative part is not divisible by (r^2 - 1)
 	// - it will not be nice to work with
 	Opaque
@@ -56,7 +65,11 @@ func (dt DigitType) String() string {
 		tString = "0"
 	case Regular:
 		tString = "R"
-	case Neighbour:
+	case Omega:
+		tString = "ω"
+	case Alpha:
+		tString = "α"
+	case Pseudoneighbour:
 		tString = "N"
 	case Opaque:
 		tString = "P"
@@ -100,8 +113,12 @@ func calcTotativeType(totative, radix uint) TotativeType {
 		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 Neighbour
+		return Pseudoneighbour
 	default:
 		return Opaque
 	}
-- 
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