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 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}?