mersenneforum.org distribution of prime numbers in special range
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 2018-10-04, 04:03 #1 miket   May 2013 32 Posts distribution of prime numbers in special range Let's start from the first prime 2, primes from 2+1 to 2^2+1 is 3,5, primes from 5+1 to 5^2+1 is 7,11,13,17,19,23 (6 primes) primes from 23+1 to 23^2+1 is 29,...523 (90 primes) primes from 523+1 to 523^2+1 is 541,...273527 (23826 primes) from wolframalpha, primepi(n^2+1)-primepi(n+1) at n=273527 get 3118310544 the sequence is 1,2,6,90,23826,3118310544 How about the next item of the sequence?
 2018-10-04, 06:32 #2 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 24·613 Posts You ask us to count primes to about 10^20, so the approximate answer is "2,220,819,602,560,918,840", well... hehe... (link) Last fiddled with by LaurV on 2018-10-04 at 06:32
 2018-10-04, 13:47 #3 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 100111011111002 Posts What is "special" about this range?
2018-10-04, 14:04   #4
Dr Sardonicus

Feb 2017
Nowhere

120058 Posts

Quote:
 Originally Posted by miket from wolframalpha, primepi(n^2+1)-primepi(n+1) at n=273527 get 3118310544 the sequence is 1,2,6,90,23826,3118310544 How about the next item of the sequence?
Doesn't wolframalpha tell you? Or does it say something like, "I'm sorry, Dave, I'm afraid I can't do that."?
:-D

Let's see here... The first nine terms of the sequence are

1 primepi(2) - primepi(1)
2 primepi(5) - primepi(2)
3 primepi(23) - primepi(5)
4 primepi(523) - primepi(23)
5 primepi(273527) - primepi(523)
6 primepi(74817019691) - primepi(273527)
7 primepi(5597586435443481735313) - primepi(74817019691)
8 primepi(31332973902260863916713294607866229791207871) - primepi(5597586435443481735313)
9 primepi(981755253559760390192770373112462804615950389612557017724898646851679425786813132352477) - primepi(31332973902260863916713294607866229791207871)

Oh, you want exact numerical values? Well, by term 7 we're beyond most tables of primes, but primepi(5597586435443481735313) might be doable in a not-unreasonable length of time without actually finding all the primes up to 5597586435443481735313.

Alas, I'm too lazy to look up the details.

You can, of course, get estimates on the growth of the sequence of primes, and on the corresponding values of primepi(). The calculation of the next few primes in the sequence may be within reach.

But absent a compelling reason, I'm not going to devote any effort to finding the primepi() values.

Last fiddled with by Dr Sardonicus on 2018-10-04 at 14:06 Reason: fixign tyops

2018-10-04, 14:22   #5
CRGreathouse

Aug 2006

3·1,993 Posts

Quote:
 Originally Posted by Dr Sardonicus Oh, you want exact numerical values? Well, by term 7 we're beyond most tables of primes, but primepi(5597586435443481735313) might be doable in a not-unreasonable length of time without actually finding all the primes up to 5597586435443481735313. Alas, I'm too lazy to look up the details.
It should take less than a CPU-day on
https://github.com/kimwalisch/primecount

A good multi-core CPU might knock it out in an hour.

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