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prime powers < X

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  • Kermit Rose
    1. number of prime powers Posted by: Werner D. Sand Theo.3.1415@web.de theo2357 Date: Tue Dec 18, 2007 4:00 pm ((PST)) Who knows a good approximate formula
    Message 1 of 2 , Dec 19, 2007
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      1. number of prime powers
      Posted by: "Werner D. Sand" Theo.3.1415@... theo2357
      Date: Tue Dec 18, 2007 4:00 pm ((PST))

      Who knows a good approximate formula for the number of prime powers up
      to x (without simple prime numbers):
      N = sum(1)(p^n <= x), p prime, n>1 ?


      *******


      Would this not be the number of primes < x
      + the number of primes < x^(1/2)
      + the number of primes < x^(1/3)
      + the number of primes < x^(1/4)
      + etc


      approximately equal to

      [ x + 2 * x^(1/2) + 3 * x^(1/3) + 4 * x^(1/4) + 5 * x^(1/5)
      + . . . ] / ln(x)




      Kermit < kermit@... >
    • Werner D. Sand
      ... up ... (1/5) ... That s allright. Can you sum it up into a closed form? WDS
      Message 2 of 2 , Dec 20, 2007
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        --- In primenumbers@yahoogroups.com, Kermit Rose <kermit@...> wrote:
        >
        >
        >
        >
        > 1. number of prime powers
        > Posted by: "Werner D. Sand" Theo.3.1415@... theo2357
        > Date: Tue Dec 18, 2007 4:00 pm ((PST))
        >
        > Who knows a good approximate formula for the number of prime powers
        up
        > to x (without simple prime numbers):
        > N = sum(1)(p^n <= x), p prime, n>1 ?
        >
        >
        > *******
        >
        >
        > Would this not be the number of primes < x
        > + the number of primes < x^(1/2)
        > + the number of primes < x^(1/3)
        > + the number of primes < x^(1/4)
        > + etc
        >
        >
        > approximately equal to
        >
        > [ x + 2 * x^(1/2) + 3 * x^(1/3) + 4 * x^(1/4) + 5 * x^
        (1/5)
        > + . . . ] / ln(x)
        >
        >
        >
        >
        > Kermit < kermit@... >


        That's allright. Can you sum it up into a closed form?
        WDS
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