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.)
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