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Old 2019-02-05, 12:15   #1
robert44444uk
 
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Jun 2003
Oxford, UK

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Default Rough number behaviour close to record gaps

I have investigated rough number behaviour close to the first instance prime gaps from 100 to 1380 to see whether any of the first instance gaps were close to nearby regions of high incidence of smooth numbers (or its converse: low incidence of rough numbers).

For this post I have constructed a graph which is based on looking at the incidence of 41-rough numbers in ranges close to where the first instance prime gaps are. A 41-rough number is a number where all prime factors are >41.

a worked example:

Take the first instance gap of length 1380 with the lower prime p(1) 1031501833130243273.

The integer range between the lower and upper prime p(2) contains g=193 41-rough integers.

I looked at n ranges of 1380 integers, commencing at p(1)+1, p(1)+2...p(1)+n and determined the numbers x of 41-rough integers, x(n1),x(n2).... in each range. I then determined the largest count x(max) and smallest count x(min) of 41-rough integers in the n ranges.

The expected number E of 41-rough integers in an integer range of 1380 is 14.50937% of 1380 = 200.223

For n = 1e6, I found x(max) and x(min) were 216 and 187

I plotted the differences between the g and E, x(max) and E and x(min) and E and express these differences as a ratio compared to E

The resulting graph shows that, although g is normally lower than E, this is not always the case, and rarely does g come anywhere close to x(min).

In the graph:.
  • the x-axis is gap/2
  • the red line represents (E-x(min))/E
  • the blue line (E-g)/E
  • the grey line (E-x(max))/E


x(min) approximates to the idea of using offsets to primorials P# which seek to find regions where there are very small numbers of P-rough numbers, hence many offset records are in integer regions where the count of rough numbers are close to and perhaps even exceed x(min)
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Old 2019-02-07, 16:41   #2
Bobby Jacobs
 
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May 2018

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That is very cool. Another thing is if there are big clusters of primes near record prime gaps. For example, the prime nonuplet 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307 is near the record prime gap from 1327 to 1361.
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