φ(n) And D(n)

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φ(n) And D(n)

Double sharp
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Joined: Sep 19 2015, 11:02 AM

Oct 13 2015, 09:37 AM #1

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: Select all

   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
wendy.krieger

Oct 13 2015, 09:58 AM #2

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
Dozens Demigod
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Joined: Sep 19 2015, 11:02 AM

Oct 13 2015, 12:14 PM #3

wendy.krieger @ Oct 13 2015, 09:58 AM wrote: 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
Dozens Demigod
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Oct 13 2015, 01:24 PM #4

Up to 240:

Code: Select all

   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
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Oct 13 2015, 01:31 PM #5

[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.[/quote] Here you go: 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.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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icarus
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Oct 13 2015, 02:48 PM #6

This comparison always has fascinated me.
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
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Oct 14 2015, 01:44 PM #7

icarus @ Oct 13 2015, 02:48 PM wrote: This comparison always has fascinated me.
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 &#966;(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
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Oct 19 2015, 01:00 PM #8

icarus @ Oct 13 2015, 02:48 PM wrote: This comparison always has fascinated me.
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
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Oct 19 2015, 04:06 PM #9

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):
icarus @ Jun 24 2011, 03:23 AM wrote: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
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Oct 20 2015, 01:56 PM #10

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 &#966;(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
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Oct 21 2015, 10:08 AM #11

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
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Oct 21 2015, 09:56 PM #12

@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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Oct 22 2015, 07:45 AM #13

icarus @ Oct 21 2015, 09:56 PM wrote: @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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Feb 4 2017, 04:34 PM #14

Come to think of it, to compute resistance, maybe we should really consider &#966;(n) - 2 instead of &#966;(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 &#966;(n) - 3, we would also admit to the club {9, 16, 36}; &#966;(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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Feb 4 2017, 05:06 PM #15

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 &#966;(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 &#966;(n) - 4 makes sense, because they are only singly even, and the bar is a little higher for them: they need &#966;(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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Nov 19 2017, 12:07 PM #16

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
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Nov 19 2017, 02:13 PM #17

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 ln³(120) / 2 ln(2) ln(3) ln(5).
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Nov 19 2017, 02:20 PM #18

wendy.krieger @ Nov 19 2017, 02:13 PM wrote: ln³(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
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Nov 19 2017, 02:34 PM #19

ln³(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 * n²/2 + ga n + gb.

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

Code: Select all

/* 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    &#59; gb = 4; gc = -18
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Double sharp
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Nov 19 2017, 02:48 PM #20

Yes, but there are only 36 regular digits in base 120: So how does the value of 44.766173 relate to that?
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wendy.krieger
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Nov 19 2017, 02:52 PM #21

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

44.766173 n² - 12.95 n + 4.

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

See, eg this thread.
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icarus
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Nov 19 2017, 09:29 PM #22

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: Select all

regulars&#91;n_&#93; &#58;= 
 Block&#91;{w }, 
 &nbsp;Sort@ToExpression@
 &nbsp; &nbsp; &nbsp;Function&#91;w, 
 &nbsp; &nbsp; &nbsp; &nbsp;StringJoin&#91;"Module&#91;{n = ", ToString@n, 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; "}, Flatten@ Table&#91;", 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; StringJoin@
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Riffle&#91;Map&#91;ToString@#1 <> "^" <> ToString@#2 & @@ # &, w&#93;,
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;" * "&#93;, ", ", Most@Flatten@Map&#91;{#, ", "} &, #&#93;, "&#93;&#93;"&#93; &@
 &nbsp; &nbsp; &nbsp; &nbsp; MapIndexed&#91;
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Function&#91;p, 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; StringJoin&#91;"{", ToString@Last@p, ", 0, Log&#91;", 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;ToString@First@p, ", n/&#40;", 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;ToString@
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; InputForm&#91;
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Times @@ Map&#91;Power @@ # &, Take&#91;w, First@#2 - 1&#93;&#93;&#93;, 
 &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;"&#41;&#93;}"&#93;&#93;@w&#91;&#91;First@#2&#93;&#93; &, w&#93;&#93;@
 &nbsp; &nbsp; &nbsp; Map&#91;{#, ToExpression&#91;"p" <> ToString@PrimePi@#&#93;} &, #&#91;&#91;All, 
 &nbsp; &nbsp; &nbsp; &nbsp; 1&#93;&#93; &#93; &@FactorInteger@n
 &nbsp;&#93;
which writes a program for each number based on its factorization:

Code: Select all

Module&#91;{n = 120}, 
 Flatten@ Table&#91;
 &nbsp; 2^p1 * 3^p2 * 5^p3, {p1, 0, Log&#91;2, n/&#40;1&#41;&#93;}, 
 &nbsp; {p2, 0, Log&#91;3, n/&#40;2^p1&#41;&#93;}, 
{p3, 0, Log&#91;5, n/&#40;2^p1*3^p2&#41;&#93;}&#93;&#93;
It could be written by a Do loop and compiled and would be greased effin' lightning:

Code: Select all

Module&#91;{k = 0, n = 120}, 
 Do&#91;
 &nbsp;Do&#91;
 &nbsp; Do&#91;k++, {p3, 0, Log&#91;5, n/&#40;2^p1*3^p2&#41;&#93;}&#93;,
 &nbsp; {p2, 0, Log&#91;3, n/&#40;2^p1&#41;&#93;}&#93;,
 &nbsp;{p1, 0, Log&#91;2, n/&#40;1&#41;&#93;}
 &nbsp;&#93;; k&#93;
(This is the counter).
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Double sharp
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Nov 19 2017, 11:51 PM #23

Ah, okay, I get it now, though Wendy might certainly have been clearer, especially about this being an approximation.
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icarus
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Nov 20 2017, 03:28 PM #24

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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