Re: [math-fun] math-fun Digest, Vol 154, Issue 36
J?rg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice.
--it seems to me, there must be some value in an area-filling curve that is the "limit" of a sequence curve1, curve2, ..., in which every curveN never self-intersects and "stays away from intersecting itself" by at least some natural distance f(N). Since in any real application, N will be finite. Possible way to make that precise: Each point of curve N is distance >= f(N) away from any other point of curveN that is not at ArcDistance <= 2*f(N). So then the question is: which area-filling curves can be manufactured in this way, and which cannot? One might argue they all can be done (proof: local surgery as needed) but perhaps not in a "nice" way (simple nice definition).
Among numbers not divisible by any prime less than p, there should be exactly an asymptotic probability of 1/p that they are divisible by p. But this seems to approach the asymptote very irregularly. Frequent case in point: Numbers relatively prime to 2*3*5 = 30 that are divisible by 7. Here are some statistics: Up through 10 there is a fraction of 0.1000000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30. Up through 100 there is a fraction of 0.1600000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30. Up through 1000 there is a fraction of 0.1433962264150943 numbers whose first prime factor is 7 among numbers relatively prime to 30. Up through 10000 there is a fraction of 0.1429643527204503 numbers whose first prime factor is 7 among numbers relatively prime to 30. Up through 100000 there is a fraction of 0.1428464279017439 numbers whose first prime factor is 7 among numbers relatively prime to 30. Up through 1000000 there is a fraction of 0.1428571428571428 numbers whose first prime factor is 7 among numbers relatively prime to 30. Why the sudden huge accuracy at 1 million? (It's a lot less accurate at 10 million.) —Dan
Well, the count can only be off by an integer. --ms On 23-Dec-15 16:15, Dan Asimov wrote:
Among numbers not divisible by any prime less than p, there should be exactly an asymptotic probability of 1/p that they are divisible by p.
But this seems to approach the asymptote very irregularly.
Frequent case in point: Numbers relatively prime to 2*3*5 = 30 that are divisible by 7.
Here are some statistics:
Up through 10 there is a fraction of 0.1000000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 100 there is a fraction of 0.1600000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000 there is a fraction of 0.1433962264150943 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 10000 there is a fraction of 0.1429643527204503 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 100000 there is a fraction of 0.1428464279017439 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000000 there is a fraction of 0.1428571428571428 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Why the sudden huge accuracy at 1 million?
(It's a lot less accurate at 10 million.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Maybe so, though 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 make 11 numbers in a row that are relatively prime to 30 without any that are divisible by 7. —Dan
On Dec 23, 2015, at 1:22 PM, Mike Speciner <ms@alum.mit.edu> wrote:
Well, the count can only be off by an integer.
--ms
On 23-Dec-15 16:15, Dan Asimov wrote:
Among numbers not divisible by any prime less than p, there should be exactly an asymptotic probability of 1/p that they are divisible by p.
But this seems to approach the asymptote very irregularly.
Frequent case in point: Numbers relatively prime to 2*3*5 = 30 that are divisible by 7.
Here are some statistics:
[deleted]
Up through 100 there is a fraction of 0.1600000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000 there is a fraction of 0.1433962264150943 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 10000 there is a fraction of 0.1429643527204503 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 100000 there is a fraction of 0.1428464279017439 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000000 there is a fraction of 0.1428571428571428 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Why the sudden huge accuracy at 1 million?
(It's a lot less accurate at 10 million.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This was carelessly added by hand:
On Dec 23, 2015, at 1:15 PM, Dan Asimov <asimov@msri.org> wrote:
Up through 10 there is a fraction of 0.1000000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
but should have been omitted.
Up through any multiple of 210 there is a fraction of exactly 1/7 numbers whose first prime factor is 7 among numbers relatively prime to 30. So for numbers not multiples of 210, the fraction will converge to 1/7, since the part played by the multiples of 210 will be more and more important compared to the fractional part of a block at the end. 1000000 happens to be close because 1000020 is a multiple of 210, and of the numbers from 1 to 20, or from 1000000 to 1000020 to 210 that are relatively prime to 30, 1 out of 5 is a multiple of 7, which is pretty close to 1 out of 7. On Wed, Dec 23, 2015 at 4:15 PM, Dan Asimov <asimov@msri.org> wrote:
Among numbers not divisible by any prime less than p, there should be exactly an asymptotic probability of 1/p that they are divisible by p.
But this seems to approach the asymptote very irregularly.
Frequent case in point: Numbers relatively prime to 2*3*5 = 30 that are divisible by 7.
Here are some statistics:
Up through 10 there is a fraction of 0.1000000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 100 there is a fraction of 0.1600000000000000 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000 there is a fraction of 0.1433962264150943 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 10000 there is a fraction of 0.1429643527204503 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 100000 there is a fraction of 0.1428464279017439 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Up through 1000000 there is a fraction of 0.1428571428571428 numbers whose first prime factor is 7 among numbers relatively prime to 30.
Why the sudden huge accuracy at 1 million?
(It's a lot less accurate at 10 million.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
Hum, maybe because 999999 is divisible by 7, 999999/7 = 142857, ? just a tought, Simon Plouffe
On 12/24/15, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hum, maybe because 999999 is divisible by 7, 999999/7 = 142857, ?
just a tought,
Simon Plouffe _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
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Andy Latto -
Dan Asimov -
Fred Lunnon -
Mike Speciner -
Simon Plouffe -
Warren D Smith