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 φ(n) And D(n), Some values
Double sharp
Posted: Oct 13 2015, 09:37 AM


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I've been curious about comparing these two values, so here's a list. It gives every integer up to 60, and then the runner-up HCNs and primorials up to 840.

I wanted it to show every integer up to 360, but what a task to do manually. I'd also have loved to get a column for φ(n)/d(n), but the alignment is confusing me to no end. So, please treat this as a call for help.
CODE

n φ d
1 1 1
2 1 2
3 2 2
4 2 3
5 4 2
6 2 4
7 6 2
8 4 4
9 6 3
10 4 4
11 10 2
12 4 6
13 12 2
14 6 4
15 8 4
16 8 5
17 16 2
18 6 6
19 18 2
20 8 6
21 12 4
22 10 4
23 22 2
24 8 8
25 20 3
26 12 4
27 18 4
28 12 6
29 28 2
30 8 8
31 30 2
32 16 6
33 20 4
34 16 4
35 24 4
36 12 9
37 36 2
38 18 4
39 24 4
40 16 8
41 40 2
42 12 8
43 42 2
44 20 6
45 24 6
46 22 4
47 46 2
48 16 10
49 42 3
50 20 6
51 32 4
52 24 6
53 52 2
54 18 8
55 40 4
56 24 8
57 36 4
58 28 4
59 58 2
60 16 12
72 24 12
84 24 12
90 24 12
96 32 12
108 36 12
120 32 16
168 48 16
180 48 18
210 48 16
240 64 20
336 96 20
360 96 24
420 96 24
480 128 24
504 144 24
600 160 24
630 144 24
660 160 24
672 192 24
720 192 30
840 192 32
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wendy.krieger
Posted: Oct 13 2015, 09:58 AM


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I take it that this is Euler Totient divided by number of divisors. Should be able to bash a rexx script to do this.

You can use a text editor to get the alignment right. I use metapad, but it depends on what form of DOS you use.
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Double sharp
Posted: Oct 13 2015, 12:14 PM


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QUOTE (wendy.krieger @ Oct 13 2015, 09:58 AM)
I take it that this is Euler Totient divided by number of divisors. Should be able to bash a rexx script to do this.

You can use a text editor to get the alignment right. I use metapad, but it depends on what form of DOS you use.

Yup, it is Euler totient over number of divisors. I would also love to have Euler totient over number of regular digits.

You're right, I didn't think of that...next time I'll be composing this sort of thing in a text editor, using a monospaced font instead.
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Double sharp
Posted: Oct 13 2015, 01:24 PM


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Up to 240:
CODE

n φ d φ/d
1 1 1 1
2 1 2 0.5
3 2 2 1
4 2 3 0.6666
5 4 2 2
6 2 4 0.5
7 6 2 3
8 4 4 1
9 6 3 2
10 4 4 1
11 10 2 5
12 4 6 0.6666
13 12 2 6
14 6 4 1.5
15 8 4 2
16 8 5 1.6
17 16 2 8
18 6 6 1
19 18 2 9
20 8 6 1.3333
21 12 4 3
22 10 4 2.5
23 22 2 11
24 8 8 1
25 20 3 6.6666
26 12 4 3
27 18 4 4.5
28 12 6 2
29 28 2 14
30 8 8 1
31 30 2 15
32 16 6 2.6666
33 20 4 5
34 16 4 4
35 24 4 6
36 12 9 1.3333
37 36 2 18
38 18 4 4.5
39 24 4 6
40 16 8 2
41 40 2 20
42 12 8 1.5
43 42 2 21
44 20 6 3.3333
45 24 6 4
46 22 4 4.5
47 46 2 23
48 16 10 1.6
49 42 3 14
50 20 6 3.3333
51 32 4 8
52 24 6 4
53 52 2 26
54 18 8 2.25
55 40 4 10
56 24 8 3
57 36 4 9
58 28 4 7
59 58 2 29
60 16 12 1.3333
61 60 2 30
62 30 4 7.5
63 36 6 6
64 32 7 4.5714
65 48 4 12
66 20 8 2.5
67 66 2 33
68 32 6 5.3333
69 44 4 11
70 24 8 3
71 70 2 35
72 24 12 2
73 72 2 36
74 36 4 9
75 40 6 6.6666
76 36 6 6
77 60 4 15
78 24 8 3
79 78 2 39
80 32 19 3.2
81 54 5 10.8
82 40 4 10
83 82 2 41
84 24 12 2
85 64 4 16
86 42 4 10.5
87 56 4 14
88 40 8 5
89 88 2 44
90 24 12 2
91 72 4 18
92 44 6 7.3333
93 60 4 15
94 46 4 11.5
95 72 4 18
96 32 12 2.6666
97 96 2 48
98 42 6 7
99 60 6 10
100 40 9 4.4444
101 100 2 50
102 32 8 4
103 102 2 51
104 48 8 6
105 48 8 6
106 52 4 13
107 106 2 53
108 36 12 3
109 108 2 54
110 40 8 5
111 72 4 18
112 48 10 4.8
113 112 2 56
114 36 8 4.5
115 88 4 22
116 56 6 9.3333
117 72 6 12
118 58 4 14.5
119 96 4 24
120 32 16 2
121 110 3 36.6666
122 60 4 15
123 80 4 20
124 60 6 10
125 100 4 25
126 36 12 3
127 126 2 63
128 64 8 8
129 84 4 21
130 48 8 6
131 130 2 65
132 40 12 3.3333
133 108 4 27
134 66 4 16.5
135 72 8 9
136 64 8 8
137 136 2 68
138 44 8 5.5
139 138 2 69
140 48 12 4
141 92 4 23
142 70 4 17.5
143 120 4 30
144 48 15 3.2
145 112 4 28
146 72 4 18
147 84 6 14
148 72 6 12
149 148 2 74
150 40 12 3.3333
151 150 2 75
152 72 8 9
153 96 6 16
154 60 8 7.5
155 120 4 30
156 48 12 4
157 156 2 78
158 78 4 19.5
159 104 4 26
160 64 12 5.3333
161 132 4 33
162 54 10 5.4
163 162 2 81
164 80 6 13.3333
165 80 8 10
166 82 4 20.5
167 166 2 83
168 48 16 3
169 156 3 52
170 64 8 8
171 108 6 18
172 84 6 14
173 172 2 86
174 56 8 7
175 120 6 20
176 80 10 8
177 116 4 29
178 88 4 22
179 178 2 89
180 48 18 2.6666
181 180 2 90
182 72 8 9
183 120 4 30
184 88 8 11
185 144 4 36
186 60 8 7.5
187 160 4 40
188 92 6 15.3333
189 108 8 13.5
190 72 8 9
191 190 2 95
192 64 14 4.5714
193 192 2 96
194 96 4 24
195 96 8 12
196 84 9 9.3333
197 196 2 98
198 60 12 5
199 198 2 99
200 80 12 6.6666
201 132 4 33
202 100 4 25
203 168 4 42
204 64 12 5.3333
205 160 4 40
206 102 4 25.5
207 132 6 22
208 96 10 9.6
209 180 4 45
210 48 16 3
211 210 2 105
212 104 6 17.3333
213 140 4 35
214 106 4 26.5
215 168 4 42
216 72 16 4.5
217 180 4 45
218 108 4 27
219 144 4 36
220 80 12 6.6666
221 192 4 48
222 72 8 9
223 222 2 111
224 96 12 8
225 120 10 12
226 112 4 28
227 226 2 113
228 72 12 6
229 228 2 114
230 88 8 11
231 120 8 15
232 112 8 14
233 232 2 116
234 72 12 6
235 184 4 46
236 116 6 19.3333
237 156 4 39
238 96 8 12
239 238 2 119
240 64 20 3.2
336 96 20 4.8
360 96 24 4
420 96 24 4
480 128 24 5.3333
504 144 24 6
600 160 24 6.6666
630 144 24 6
660 160 24 6.6666
672 192 24 8
720 192 30 6.4
840 192 32 6

The numbers that give a value of φ(n)/d(n) less than or equal to 2 are {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 60, 72, 84, 90, 120}.

If we move the limit down to 1.5 (3/2), it becomes {1, 2, 3, 4, 6, 8, 10, 12, 14, 18, 20, 24, 30, 36, 42, 60}.

If we move it down further to 1.333... (4/3), we get {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 60}. An interesting sequence! Given how {18, 20} are comparable in scale, this may be the actual limit (with a slight totative dominance) for whether a base mostly resists us or helps us.

If we move it down to 1, we get {1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30} and no others. I am not convinced completely about this sequence, because {20, 60, (120)} have appeared in pretty advanced societies, and therefore seem to be workable. The 4/3 limit seems better, though I'm not sure if {120} is really as bad as this valuation paints it as (although I think everything else beyond {60} is surely unusable).
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Kodegadulo
Posted: Oct 13 2015, 01:31 PM


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QUOTE (Double sharp @ Oct 13 2015, 09:37 AM)
I've been curious about comparing these two values, so here's a list. It gives every integer up to 60, and then the runner-up HCNs and primorials up to 840.

I wanted it to show every integer up to 360, but what a task to do manually. ...So, please treat this as a call for help.

Here you go:
nφd
111
212
322
423
542
624
762
844
963
1044
11102
1246
13122
1464
1584
1685
17162
1866
19182
2086
21124
22104
23222
2488
25203
26124
27184
28126
29282
3088
31302
32166
33204
34164
35244
36129
37362
38184
39244
40168
41402
42128
43422
44206
45246
46224
47462
481610
49423
50206
51324
52246
53522
54188
55404
56248
57364
58284
59582
601612
722412
842412
902412
963212
1083612
1203216
1684816
1804818
2104816
2406420
3369620
3609624
4209624
48012824
50414424
60016024
63014424
66016024
67219224
72019230
84019232


Do a Quote to see the BBNcode/HTML. I used a little macro recording in notepad++ to format one table row line and then just repeated it (re-recording it when I got to 3-digit numbers). You could certainly doctor any script to output the same codes.
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icarus
Posted: Oct 13 2015, 02:48 PM


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This comparison always has fascinated me.

QUOTE
If we move it down to 1, we get {1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30} and no others.


Check this out: OEIS A020490. I added code there this morning: Select[Range@ 1000000, EulerPhi@ # <= DivisorSigma[0, #] &] , and once I can track it down I'll add a reference I know about. It's a very interesting sequence.

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Double sharp
Posted: Oct 14 2015, 01:44 PM


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QUOTE (icarus @ Oct 13 2015, 02:48 PM)
This comparison always has fascinated me.

QUOTE
If we move it down to 1, we get {1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30} and no others.


Check this out: OEIS A020490. I added code there this morning: Select[Range@ 1000000, EulerPhi@ # <= DivisorSigma[0, #] &] , and once I can track it down I'll add a reference I know about. It's a very interesting sequence.

Yes, it's the same sequence. However, given the use of {20} by the Mayans and {60} by the Sumerians, I think the limit might be just a little higher than 1. To admit these two bases, we need to raise the limit to at most a 3:4 ratio between divisors and totatives, which adds {20, 36, 60} to the list. I'm somewhat convinced by this as {60} does appear to be very useful, {20} and {18} are comparable in scale (having the same number of divisors and opaque totatives), and removing the transparent non-unitary totatives from {36}'s count (5, 7, and 35) brings it to totative-divisor parity.

OTOH, {120} has φ(120)/d(120) = 2. So I am not quite sure if going up to {120} is worth it, because all you gain from {60} are single-digit eighths (a significant plus if you use 6:10 and 12:10; not very significant if you use pure sexagesimal or centovigesimal), and better neighbour relationships (but they only ameliorate four totatives out of thirty-two, so that the ratio remaining is still 1.75).

Perhaps regular digits are significant too. Instinctively it feels wrong to me to count semidivisors equally with divisors, but I don't know what a fair valuation would be, so I have to try it:

Senary (6) - 5 regular, 2 totatives: ratio is 2.5
Octal (8) - 4 regular, 4 totatives: ratio is 1
Decimal (10) - 4 regular, 4 totatives: ratio is 1
Duodecimal (12) - 8 regular, 4 totatives: ratio is 2
Tetradecimal (14) - 6 regular, 6 totatives: ratio is 1
Hexadecimal (16) - 5 regular, 8 totatives: ratio is 0.625
Octodecimal (18) - 10 regular, 6 totatives: ratio is 1.666...
Vigesimal (20) - 8 regular, 8 totatives: ratio is 1
Tetravigesimal (24) - 11 regular, 8 totatives: ratio is 1.375
Octovigesimal (28) - 8 regular, 12 totatives: ratio is 0.666...
Trigesimal (30) - 18 regular, 8 totatives: ratio is 2.25
Hexatrigesimal (36) - 14 regular, 12 totatives: ratio is 1.166...
Duoquadragesimal (42) - 19 regular, 12 totatives: ratio is 1.583...
Octoquadragesimal (48) - 15 regular, 16 totatives: ratio is 0.9375
Sexagesimal (60) - 26 regular, 16 totatives: ratio is 1.625
Septuagesimal (70) - 18 regular, 24 totatives: ratio is 0.75
Duoseptuagesimal (72) - 18 regular, 24 totatives: ratio is 0.75
Octogesimal (80) - 14 regular, 32 totatives: ratio is 0.4375
Tetroctogesimal (84) - 28 regular, 24 totatives: ratio is 1.166...
Nonogesimal (90) - 32 regular, 24 totatives: ratio is 1.333...
Hexanonogesimal (96) - 20 regular, 32 totatives: ratio is 0.625
Centesimal (100) - 15 regular, 40 totatives: ratio is 0.375
Centoctonary (108) - 21 regular, 36 totatives: ratio is 0.583...
Centoduodecimal (112) - 14 regular, 48 totatives: ratio is 0.2916...
Centovigesimal (120) - 36 regular, 32 totatives: ratio is 1.125...
Centotetraquadragesimal (144) - 23 regular, 48 totatives: ratio is 0.47916...
Duocentodecimal (210) - 50 regular, 48 totatives: ratio is 1.0416...
Duocentoquadragesimal (240) - 51 regular, 64 totatives: ratio is 0.796875
Trecentosexagesimal (360) - 61 regular, 96 totatives: ratio is 0.635416...
Septingentovigesimal (720) - 76 regular, 192 totatives: ratio is 0.39583...

(I'd have loved to include 2310 and 2520, but couldn't find counts of regulars.)

The 1-and-above club in this list (ignoring the really small bases below 5 that get in simply for having very few totatives) is {6, 8, 10, 12, 14, 18, 20, 24, 30, 36, 42, 60, 84, 90, 120, 210}. But I think that this equal valuation is biased in favour of primorials, and does not consider that regulars like 512 in {720} are of no practical help whatsoever (they're too "rich", dividing too high a power of the base).

So I don't really know of a measure that seems to reflect {120} as an "island of stability" in the sea of instability beyond 36, as I thought it would be. ({60} is assuredly an island.) Perhaps the added totative resistance is really not worth it.

Is this because this measurement is geared towards pure radices instead of things like {12:10}?
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Double sharp
Posted: Oct 19 2015, 01:00 PM


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QUOTE (icarus @ Oct 13 2015, 02:48 PM)
This comparison always has fascinated me.

QUOTE
If we move it down to 1, we get {1, 2, 3, 4, 6, 8, 10, 12, 18, 24, 30} and no others.

One thing that I find particularly interesting about this list is that the only bases here that have a gap in their prime factorization are {3, 10}. If we add {20, 36, 60}, in effect raising the maximum acceptable totative:divisor ratio to 4/3, then the list becomes {3, 10, 20}. {3} is probably in there due to what I call "edge effects": in the very low bases, there are so few digits that you can get oddities like all but one digit being regular ({4, 6}), no non-trivial totatives and divisors ({2}), and primes with no opaque digits ({2, 3, 5}). This strange behaviour seems to stop at {7}, at which a new human-scale regime appears to start and continue to {15} (excepting {13}).

So, if we discount {3} for being too small for today's society, and odd at that, it does appear than {20}, and {10} even more so, are really exceptional bases, that can still compete in spite of their less efficient prime decomposition.
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Double sharp
Posted: Oct 19 2015, 04:06 PM


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The sequence A072938 shows the HCNs which are half of the next HCN: {1, 2, 6, 12, 60, 360, 2520} and no others. (Here's a proof in German.)

The first five appear in our sequence with totative:divisor ratio set to 4:3, but the last two are huge, at {360, 2520}, only to add the prime powers {8, 9, 7}. {360} concentrates extra weight on previously unrepresented powers of already existing primes, for no benefit in reducing the totient ratio, so that the totatives end up swamping the divisors. And {2520} does add another prime factor 7, but is even more swamped with totatives due to its huge size.

Now I think I see why {120} does so well despite having a large totative:divisor ratio! It handles 8 as a divisor and 7 as a keen omega-totative, only lacking the 9 contributed by {360}! It's acting like the little brother of the absent {2520}, only failing to handle ninths in one place! And in exchange it handles the next prime {11} gracefully too, as well as the second prime after that, {17}! (The missing prime {13} is not even maximal, being handled by the cube-omega!) The wonderful sevenths seem to be a good trade-off for the slightly worse ninths when comparing {120} against {360}!

The drawback of this is that while I am very sure now that {120} truly does rank in usefulness equally with the numbers with at most 4/3 as many totatives as divisors - {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 60} - I do not really have a legitimate mathematical formula that gives this valuation of {120} yet. All I have to support my case here is how {120} handles sevenths and eighths pretty well at an incredible twenty-first of the scale of {2520}, the first SHCN to handle them both as divisors. It really seems to be one of the last islands of usefulness, if not the last, thanks to it being the only low ({5040} and below) SHCN that has only composite neighbours. As Icarus said, comparing the most useful small SHCN bases {2, 6, 12, 60, 120}, with a brief look at decimal (emphasis mine):

QUOTE (icarus @ Jun 24 2011, 03:23 AM)
Sixty is more regular-digit dense (26/60 vs. 36/120) but Dozenal is even more so (8/12) and decimal surprisingly strong (6/10). Base six is stronger (5/6) and base 2 supreme (2/2) but these small bases suffer from the length of their numbers for common, small quantities and the monotony of their small digit ranges. I think a "clean" quarter is very useful and hard to surrender once enjoyed, so another mark for twelve vs. six. I prefer sixty over twelve in examining other large bases because sixty is five smooth and thus has greater power to resolve numbers. Twelve feels encumbering compared to sixty, both in magnitude and the three-smooth vs. five smooth limitation. Twelfty would give us clean eighths, limited ability to wield sevenths, elevenths, and seventeenths (via omega and alpha properties), a brief thirteenth, and intuitive divisibility rules for 6 of 7 of the smallest primes. It has alignments at a compact enough scale where it has special properties no other number offers. Once you're larger, there is too much junk in the bag for additional resolution to matter, and if smaller, a shorter range of resolution is offered. Twelfty really is a fascinating base. (I still prefer sixty, but now outside of the greater density of its regular digits, bereft of any meaningful omegas or alphas it can't hold a candle to the broad range for twelfty).
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Double sharp
Posted: Oct 20 2015, 01:56 PM


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Another totative-related comparison I've recently been thinking about is the incidence of opaque composite totatives, like sexagesimal 49. After having done some work in sexagesimal with pen and paper, I find myself very often noting 49 as a prime, before realizing that it isn't. (Even though it isn't, it is a sexagesimal totative and so primes can end in "49", such as "1'49", decimal 109.)

The bases that lack opaque composite totatives are {2-10, 12, 15, 18, 24, 30}. I'm not surprised by the low range {2-10} as there simply are so few digits (hmm, maybe {2-10} is the "extended" low range while {2-6} is the main low range?) that even inefficient prime decompositions like {5} and {7} still work thanks to the very composite neighbours. After that, it's only the multiples of six, which work so well to govern the primes until {24}, just below 52 (the square of the first unrepresented senary prime), at which 5 steps into factorization for {30} and no more bases. I am really amazed by pentadecimal's performance here, and while I'm not sure what the best base overall is, I am now almost totally convinced that the best odd base is {15}.

An added bonus of the mid-scale bases {18, 24, 30} here is that not only do they lack composite totatives, they also have φ(n) = d(n) (the main subject of this thread), and their abbreviated multiplication tables have only regular product lines (as their first totatives - 5, 5, and 7 respectively - are above their square roots).

I must say that my opinion of trigesimal has now raised significantly! I've got to experiment with it to see if I think the loss of a clean quarter over sexagesimal is worth its smaller size, totative-divisor parity, and lack of composite totatives. I'm also impressed by tetravigesimal, as it has 5 as a base-24 alpha Wieferich prime. Octodecimal I am less impressed by, as it does not seem to offer important added advantages over duodecimal that would offset its size: but I will investigate it further nonetheless.
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wendy.krieger
Posted: Oct 21 2015, 10:08 AM


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16 and 18 are unique in having a perimeter equal to the area (4,4) and (3,6), these being regular tilings, but that's the same formula.

2,4,6,10,12, and 18 are the only bases to have prime decades, such as decimal 101, 103, 107, 109, and dozenal 81, 85, 87, 8E. The coprimes of the form 7,9,2,X in base 18 are also primes (this is the second such run). Base 30 has prime decades if one supposes a fully symmetric digits (-14 to +15), there are eight primes within 15 of 1230.

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icarus
Posted: Oct 21 2015, 09:56 PM


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@Double Sharp,

I feel your pain, regarding the sexagesimal totative 7^2. It is a "pseudoprime" that way, because our "sieve" (sexagesimal) passes it as perhaps prime because it is coprime to 60 and no other composite digit is coprime to 60. {{1, 7, 11, 13}, {17, 19, 23, 29}, {31, 37, 41, 43}, {47, [49], 53, 59}} seems so complete. I used to consider it a major thing that a base had a composite totative but all bases greater than 30 have at least one. Then I considered the base-complements of the totatives {1, 59}, {7, 53}, etc., that there must be something up with 11. In base 360 the lineups seem to indicate that composite totatives have composite complements - until they don't.
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Double sharp
Posted: Oct 22 2015, 07:45 AM


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QUOTE (icarus @ Oct 21 2015, 09:56 PM)
@Double Sharp,

I feel your pain, regarding the sexagesimal totative 7^2. It is a "pseudoprime" that way, because our "sieve" (sexagesimal) passes it as perhaps prime because it is coprime to 60 and no other composite digit is coprime to 60. {{1, 7, 11, 13}, {17, 19, 23, 29}, {31, 37, 41, 43}, {47, [49], 53, 59}} seems so complete. I used to consider it a major thing that a base had a composite totative but all bases greater than 30 have at least one. Then I considered the base-complements of the totatives {1, 59}, {7, 53}, etc., that there must be something up with 11. In base 360 the lineups seem to indicate that composite totatives have composite complements - until they don't.

In base 360 the pair {143, 217} are complements and both are totatives to 360. However, both are composite: 143 = 11 * 13 and 217 = 7 * 31. This is the only such pair in base 360.

The only bases where all opaque totatives are primes are {2-10, 12, 15, 18, 24, 30}, as previously mentioned. If we restrict this further to all totatives (even if they are transparent), the only bases with only prime or unitary totatives are {2, 3, 4, 6, 8, 12, 18, 24, 30}. I feel this is a little unfair to decimal as it is disqualified by its omega, which is surely transparent; thus I'd add it in parentheses, giving {2, 3, 4, 6, 8, (10), 12, 18, 24, 30}. This counts omega as transparent but not neighbour-factors.
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Double sharp
Posted: Feb 4 2017, 04:34 PM


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Come to think of it, to compute resistance, maybe we should really consider φ(n) - 2 instead of φ(n). The reason is that 1 never gives any resistance at all, and that the procedure for omega products is the same in any base and never requires much memorisation beyond addition. (Decrement the other multiplicand by one to give the first digit, and then pick the second digit so that they sum to give the omega again.)

By that metric, the following bases have at least as much leverage as resistance:

{(1), 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 30}

which seems to accord better with how 14 and 20 seem about equally functional to 10 and 18 respectively.

If we looked at φ(n) - 3, we would also admit to the club {9, 16, 36}; φ(n) - 4 admits also {7, 15, 42, 60}. The latter seems to accord best with my experience: 11 and 13 sneak into the human-scale range not so much because they're easy as because they're small enough that the chore of memorising the multiplication table at least has an end in sight.
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Double sharp
Posted: Feb 4 2017, 05:06 PM


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Which makes me think of another thing: what leverage is this function computing? You see, for all that the multiplication table of base thirty might be highly patterned, it would be insane to try to memorise it. (I confess to having tried to do it. In my defence, I expected to fail, and I did fail.)

And what about rote memorisation? If the number of facts is within the range that memorising all of them is reasonable, then there's not so much difference. The difference seems to only come in the mid-scale where there is a difference, including the low range {16, 18, 20} which overlaps with the top of the human scale in its methods.

So this seems to, more than anything else, simply compute leverage for bases that can use the reciprocal divisor method. Bases 15 and below don't really need this help. If you use reciprocal divisors, the neighbours of half the base are also pretty easy: they are really only one multiplicative step (the halving), so the stipulation that d(n) be at least φ(n) - 4 seems to make some sense. I think this is why Icarus has said before that 60 is semipractical in a way that 120 and 360 really aren't.

So when this metric seems to praise {24, 30, 36, 42, 60}, we have to think of it in terms of reciprocal divisors. Indeed I find that they are all pretty good for it, but it should be noted that {30, 42} don't get much of the reason why φ(n) - 4 makes sense, because they are only singly even, and the bar is a little higher for them: they need φ(n) - 2 in the mid-scale. Thirty scrapes past but forty-two doesn't. So {24, 30, 36, 60} seem to be recommended. So now all I've done is recreate the mid-scale list that you get if you say that you want no more than four-thirds as many totatives as divisors.

Oh well. It's good to have confirmation, at least.

(Yay, 800 posts!)
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Double sharp
Posted: Nov 19 2017, 12:07 PM


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Also, perhaps we might want to consider counting regulars instead of divisors, in which case we have regular parity or dominance over other digits only at {1, 2, 3, 4, 6, 8, 10, 12, 18, 30}. Which is the same list as the first, except without 24, and is A275581 in the OEIS (submitted by icarus).
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wendy.krieger
Posted: Nov 19 2017, 02:13 PM


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The number of regulars under a given base, is not really governed by the signature, but by some form

ln^n(B )/(n-1)! ln(p_1)ln(p_2) ... ln(p_n)

eg ln(120) / 2 ln(2) ln(3) ln(5).
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Double sharp
Posted: Nov 19 2017, 02:20 PM


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QUOTE (wendy.krieger @ Nov 19 2017, 02:13 PM)
ln(120) / 2 ln(2) ln(3) ln(5)

Which evaluates to about 5.859 if ln^3 is read as ln (120^3), and about 0.183 if ln^3 means ln ln ln (which it probably doesn't), and there are clearly more 5-smooth numbers under 120 than that. So what exactly did you mean here?
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wendy.krieger
Posted: Nov 19 2017, 02:34 PM


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ln(x) is (ln(x))^3. It comes to 44.766173.

You consider the regulars in logrithmetic space, like 2, 3, 5 axis. Each regular occupies a cell, and you are drawing lines around the regulars.

This is code from my regulars script, that looks compares a set of regulars to some b^n, as formulae: mantissa = digit stream regardless of radix, so 2 = 2,0 = 2,0,0 counted as one entity.

new regulars mantissa in ripple n, g0 * n - gc
total regulars mantissa to ripple n g0 * n/2 + ga n + gb.

It's only evaluated for bases with three prime divisors.

CODE

/* base K A B C */
b = 120; g0 = 89.53234771; ga = -12.95; gb = 4; gc = -57
b = 60; g0 = 56.00270746; ga = -4.7 ; gb = 2; gc = -32
b = 30; g0 = 32.10342492; ga = 0 ; gb = 1; gc = -15
b = 70; g0 = 35.32506234; ga = 0 ; gb = 1; gc = -18
b = 42; g0 = 35.237901268; ga = 0 ; gb = 1; gc = -18
b = 84; g0 = 58.702782073; ga = -4.5 ; gb = 0; gc = -34
b = 168; g0 = 90.787385076; ga = -12.25 ; gb = 4; gc = -58
b = 105; g0 = 29.297230883; ga = 0 ; gb = 4; gc = -18
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Double sharp
Posted: Nov 19 2017, 02:48 PM


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Yes, but there are only 36 regular digits in base 120:

Regular Digits gr

50 51 52
  20 21 22 23 24 25 26
30 1 2 4 8 16 32 64
31 3 6 12 24 48 96  
32 9 18 36 72      
33 27 54 108        
34 81            
  20 21 22 23 24
30 5 10 20 40 80
31 15 30 60 120  
32 45 90      
  20 21 22
30 25 50 100
31 75    

So how does the value of 44.766173 relate to that?
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wendy.krieger
Posted: Nov 19 2017, 02:52 PM


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The actual number of regulars between consecutive powers of 120, is

44.766173 n - 12.95 n + 4.

This value is to within \pm 2 up to n=30.

See, eg this thread.
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icarus
Posted: Nov 19 2017, 09:29 PM


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I did a really massive study of regular numbers. Wendy is approximating them, which is okay.

An algorithm that generates or counts regular numbers does function much like the divisor counting function. It is helpful to look at the former first.

The divisor counting function d(n) = (e_1 + 1) * (e_2 + 1) * ... (e_k + 1) (i.e., a tensor product), where the standard form prime decomposition is n = p_1^e_1 * p_2^e_2 * ... * p_k^e^k. In this case, the function produces an orthogonal figure similar to an m-dimensional parallelepiped, with m = omega(n), i.e., the number of distinct prime divisors of n. (See OEIS 275055).

The regular counting function r(n) is similar but not so easily generated this way, because of the number n acting as a bound. The charts Double sharp posted are a good way to look at it. We made m axes, one for each distinct prime divisor, and include all powers of p from 0 up to floor(log_p(n)). We make a matrix as we do for the divisors, except we don't right any numbers r > n, meaning that the table will have an irregular shape similar to a "right-simplex" in m dimensions. For squarefree semiprimes this will look roughly triangular, with the "hypotenuse" a slightly convex, ragged diagonal edge. (See OEIS 275280). Thus the shape is amenable to some analysis that Wendy suggests.

I've written an algorithm that computes regulars this way and it is the very most efficient method. (See OEIS A244052, function f):

CODE

regulars[n_] :=
Block[{w },
 Sort@ToExpression@
     Function[w,
       StringJoin["Module[{n = ", ToString@n,
          "}, Flatten@ Table[",
          StringJoin@
           Riffle[Map[ToString@#1 <> "^" <> ToString@#2 & @@ # &, w],
             " * "], ", ", Most@Flatten@Map[{#, ", "} &, #], "]]"] &@
        MapIndexed[
         Function[p,
            StringJoin["{", ToString@Last@p, ", 0, Log[",
             ToString@First@p, ", n/(",
             ToString@
              InputForm[
               Times @@ Map[Power @@ # &, Take[w, First@#2 - 1]]],
             ")]}"]]@w[[First@#2]] &, w]]@
      Map[{#, ToExpression["p" <> ToString@PrimePi@#]} &, #[[All,
        1]] ] &@FactorInteger@n
 ]


which writes a program for each number based on its factorization:

CODE

Module[{n = 120},
Flatten@ Table[
  2^p1 * 3^p2 * 5^p3, {p1, 0, Log[2, n/(1)]},
  {p2, 0, Log[3, n/(2^p1)]},
{p3, 0, Log[5, n/(2^p1*3^p2)]}]]


It could be written by a Do loop and compiled and would be greased effin' lightning:

CODE

Module[{k = 0, n = 120},
Do[
 Do[
  Do[k++, {p3, 0, Log[5, n/(2^p1*3^p2)]}],
  {p2, 0, Log[3, n/(2^p1)]}],
 {p1, 0, Log[2, n/(1)]}
 ]; k]


(This is the counter).
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Double sharp
Posted: Nov 19 2017, 11:51 PM


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Ah, okay, I get it now, though Wendy might certainly have been clearer, especially about this being an approximation.
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icarus
Posted: Nov 20 2017, 03:28 PM


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It may be keen, and perhaps there is a way via calculus to compute a "volume", however the bound by magnitude of n requires that the curve that is the "hyper-hypotenuse" ( hypertenuse? ; ) ) must include only full "cells." This non-discrete method will tend to overestimate regulars. For very large numbers this might be acceptable. I have computed the regular counting function A010846 for numbers as great as primorial p_15# (which for me takes a workday) and a solid range of 32,000,000 (I am heading for p_9# and it will take months) in order to generate data for sequences A292867, A293555, etc. and refine the conjectures therein. It would be interesting to have a non-discrete or analytical estimation method.

Where the divisor counting function has n as a boundary by nature (no number greater than n may divide n) the regular counting function has n as a less-"natural" boundary. The series of regular numbers in base n is infinite; we select n not arbitrarily, but it does "cut off" the sequence. The tensor R of regulars is the same for numbers with the same "core" (squarefree root); that of n = {6, 12, 18, 24, 36, ...} is R_6. The regular counting function merely "cuts off" a portion of R_6 at n in different places. In this case, only squarefree numbers (A005117) have unique R. Note the same is true for d(n), since this function cuts off the axes at the last power of p that divides n. R_6 is cut off at 2^1 and 3^1, and R_12 at 2^2 and 3^1, whereas r(6) cuts R_6 at any number in the tensor greater than 6, r(12) at any number in the tensor greater than 12. This shows that r(n) >= d(n) and divisors a subset of the regulars of n. It also implies that the number of semidivisors (i.e., A243822(n)) for numbers n for which omega(n) > 1 eventually swamp d(n) as n increases, and that divisors are a vanishingly small subset of the regulars of such n as n gets large.
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