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Two prime factors formula?

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  • Werner D. Sand
    Does anyone know an approximate formula (according to x/ln x) for the number of positive integers up to x which are the product of exactly 2 prime factors:
    Message 1 of 4 , Jan 31, 2007
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      Does anyone know an approximate formula (according to x/ln x) for the
      number of positive integers up to x which are the product of exactly 2
      prime factors: 4,6,9,10,14,15,21,22…? (NOT: 2 different prime factors
      as in 12, 36,225…). Perhaps a general formula for n prime factors?
    • mikeoakes2@aol.com
      An integer with exactly k prime factors is somethimes called a k-almost-prime. Here s a link:- http://mathworld.wolfram.com/AlmostPrime.html -Mike Oakes ...
      Message 2 of 4 , Jan 31, 2007
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        An integer with exactly k prime factors is somethimes called a
        k-almost-prime.

        Here's a link:-
        http://mathworld.wolfram.com/AlmostPrime.html

        -Mike Oakes


        -----Original Message-----
        From: Theo.3.1415@...
        To: primenumbers@yahoogroups.com
        Sent: Wed, 31 Jan 2007 9.13AM
        Subject: [PrimeNumbers] Two prime factors formula?

        Does anyone know an approximate formula (according to x/ln x) for the
        number of positive integers up to x which are the product of exactly 2
        prime factors: 4,6,9,10,14,15,21,22…? (NOT: 2 different prime factors
        as in 12, 36,225…). Perhaps a general formula for n prime factors?
      • Werner D. Sand
        Perfect answer! Thank you, Mike. ... the ... exactly 2 ... factors
        Message 3 of 4 , Jan 31, 2007
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          Perfect answer! Thank you, Mike.




          --- In primenumbers@yahoogroups.com, mikeoakes2@... wrote:
          >
          > An integer with exactly k prime factors is somethimes called a
          > k-almost-prime.
          >
          > Here's a link:-
          > http://mathworld.wolfram.com/AlmostPrime.html
          >
          > -Mike Oakes
          >
          >
          > -----Original Message-----
          > From: Theo.3.1415@...
          > To: primenumbers@yahoogroups.com
          > Sent: Wed, 31 Jan 2007 9.13AM
          > Subject: [PrimeNumbers] Two prime factors formula?
          >
          > Does anyone know an approximate formula (according to x/ln x) for
          the
          > number of positive integers up to x which are the product of
          exactly 2
          > prime factors: 4,6,9,10,14,15,21,22…? (NOT: 2 different prime
          factors
          > as in 12, 36,225…). Perhaps a general formula for n prime factors?
          >
        • Werner D. Sand
          I wouldn t call sum(something) a formula, but a calculating instruction. For example pi(x) = sum(1)(2,3,5…x) looks great, but is only an instruction to count
          Message 4 of 4 , Feb 2, 2007
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            I wouldn't call sum(something) a formula, but a calculating
            instruction. For example pi(x) = sum(1)(2,3,5…x) looks great, but is
            only an instruction to count primes up to x, while the formula is
            pi(x) ~ x/ln x. Thus pi(2)(x) = sum (pi(x/p i)-i+1) (i=1…pi(x^(1/2))
            looks very spectacular, but is only the description of counting the
            semiprimes up to x. The formula for it might be
            pi(2)(x) ~ n/(e ln ln x) or the like, perhaps someone knows a better
            one.

            Werner







            --- In primenumbers@yahoogroups.com, "Werner D. Sand"
            <Theo.3.1415@...> wrote:
            >
            > Perfect answer! Thank you, Mike.
            >
            >
            >
            >
            > --- In primenumbers@yahoogroups.com, mikeoakes2@ wrote:
            > >
            > > An integer with exactly k prime factors is somethimes called a
            > > k-almost-prime.
            > >
            > > Here's a link:-
            > > http://mathworld.wolfram.com/AlmostPrime.html
            > >
            > > -Mike Oakes
            > >
            > >
            > > -----Original Message-----
            > > From: Theo.3.1415@
            > > To: primenumbers@yahoogroups.com
            > > Sent: Wed, 31 Jan 2007 9.13AM
            > > Subject: [PrimeNumbers] Two prime factors formula?
            > >
            > > Does anyone know an approximate formula (according to x/ln x) for
            > the
            > > number of positive integers up to x which are the product of
            > exactly 2
            > > prime factors: 4,6,9,10,14,15,21,22…? (NOT: 2 different prime
            > factors
            > > as in 12, 36,225…). Perhaps a general formula for n prime
            factors?
            > >
            >
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