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Prime Counting Function

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  • William F Sindelar
    Hi Everybody Finally fixed the computer,so I would like to belatedly thank Jack Brennen, Phil Carmody, Joe McClean and Chris Nash for their comments on my
    Message 1 of 3 , Aug 29, 2001
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      Hi Everybody
      Finally fixed the computer,so I would like to belatedly thank Jack
      Brennen, Phil Carmody, Joe McClean and Chris Nash for their comments on
      my Sierpinski and Little Fermat posts. Gentlemen, I do appreciate your
      taking the time to answer questions I should have been able to answer for
      myself by a more diligent search of the web. I do try but sometimes find
      it frustratingly difficult.
      For example I’ve noticed that for certain primes P, the number of
      composite ODD integers up to and including P is evenly divisible by the
      number of primes up to and including P. The same for EVEN integers. Put
      in different words, the number of ODDS or EVENS equals an integer N times
      the number of primes. An approximation to N is the integer part of (LN(P)
      – 2)/2 for odds and the integer part of (LN(P) – 1)/2 for evens.
      Clearly this pertains to the Nth Prime and the Prime Counting Function
      categories and surely someone investigating these fields must have also
      noticed this, but I’ve had no luck finding any references. Maybe it’s so
      trivial it’s not worth mentioning. Here are some specifics for anyone
      interested:
      A represents the number of consecutive primes that exist up to and
      including P, labeling the prime 2 as number 1.
      B represents the number of ODD COMPOSITE integers up to and including P,
      labeling the odd integer 1 as number 1. B = ((P+1) / 2) - A
      C represents the number of EVEN integers up to P, labeling the even
      integer 2 as number 1. C = (P-1) / 2
      RO represents the ratio B / A
      RE represents the ratio C / A
      From a list of 100,000 consecutive primes which I downloaded and assume
      to be accurate, I sifted out the following data according to the criteria
      that RO and RE be integers:
      For primes having integer RO’s:
      Primes 3, 5, 1091, 8423, 64579, 64609, 64709, 481043, 481067
      A’s 2, 3, 182, 1053, 6458, 6461, 6471, 40087, 40089
      B’s 0, 0, 364, 3159, 25832, 25844, 25884, 200435, 200445
      RO’s 0, 0, 2, 3, 4, 4, 4, 5, 5
      For primes having integer RE’s:
      Primes 11, 13, 1087, 64591, 64601, 64661
      A’s 5, 6, 181, 6459, 6460, 6466
      C’s 5, 6, 543, 32295, 32300, 32330
      RE’s 1, 1, 3, 5, 5, 5
      Is it possible to prove that the supply of such P’s is infinite? Is it
      possible to find a P such that an integer RO = an integer RE? Are all
      integer RE’s odd? And a lot of other questions. I’m hoping someone having
      access to a larger database will extend my range. I ended with the prime
      1,299,827 which is prime number 100,008. I’d appreciate any comments and
      thanks everyone.
      Bill Sindelar
    • William F Sindelar
      Hi Everybody Mea Maxima Culpa. Please make the following correction to my post Prime Counting Function . In the last paragraph instead of the question Is it
      Message 2 of 3 , Aug 30, 2001
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        Hi Everybody
        Mea Maxima Culpa. Please make the following correction to my post "Prime
        Counting Function". In the last paragraph instead of the question "Is it
        possible to find a P such that an integer RO = an integer RE?" I should
        have asked "Is it possible to find a P such that A evenly divides both B
        and C?" I think it makes it a bit more understandable as to what I mean.
        My abject apology. Thanks again everyone.
        Bill Sindelar
      • Anton
        According to Prime Number Theorom a) pi_a(n) ~ Li(n) above is the improvement over b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336 For those who cannot compute the
        Message 3 of 3 , Sep 7, 2005
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          According to Prime Number Theorom

          a) pi_a(n) ~ Li(n)

          above is the improvement over

          b) pi_b(n) ~ n/(ln(n)+B) with B = -1.08336

          For those who cannot compute the logarithmic integral Li(n) the
          following counting functions yields better results than b)

          c) pi_c(n) ~ n{1 +0.5x +sqrt[x^2/4 -x-1 -1/(4x^2)]}/(2 x)
          where x=ln(n)

          let n=10^k below the differences to the actual pi(10^k)

          k ; a) ; b) ; c)

          3 ; 9 ; 3 ; 9
          6 ; 129 ; 43 ; 105
          9 ; 1700 ; 69207 ; 312
          12 ; 3.8 10^4 ; 6.1 10^7 ; -1.1 10^5
          15 ; 1.1 10^6 ; 4.6 10^10 ; -2.5 10^7
          18 ; 2.2 10^7 ; 3.5 10^13 ; -2.5 10^8
          21 ; 5.9 10^8 ; 2.7 10^16 ; -4.1 10^12
          23 ; 7.2 10^9 ; 2.3 10^18 ; -4.3 10^14

          pi_a(n) ~ Li(n) is still the best

          Has any one seen the approximation pi_c() before

          I discovered it by investigating the number of primes
          between a^b and a^(b+1)

          regards
          Anton
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