> Here is my rewrite of the formula (requires LaTeX):
>
> \documentclass{article}
> \begin{document}
> \thispagestyle{empty}\noindent
> Let $P$ be the set of primes $p$ such that $p^2<n$,
> and let $m=\left| P\right|$.
> Then the number of semiprimes less than or equal to $n$ is given by
> $$\sum_{p\in P} \left(\pi(n/p)-\pi(p)\right)=
> \sum_{p\in P} \pi(n/p)- \frac{m(m-1)}2.$$
> \end{document}
I think you need to say p^2<=n rather than <n .
Otherwise it fails for semiprimes n which are
perfect squares.
----- Original Message -----
From: John Randall <[EMAIL PROTECTED]>
Date: Saturday, April 5, 2008 13:18
Subject: Re: [Jgeneral] How readable is J?
To: General forum <[email protected]>
> Roger Hui wrote:
> > Well, it can not be exactly the same formula because
> > the J one finds semiprimes less than n while the
> > MathWorld one finds those less than or equal to n .
> > I derived the J computation before I saw the MathWorld
> > one and saw that they are very similar.
> >
>
> The difference is pi versus plt: otherwise I think they are the same.
>
> > As a mathematician, do you find the MathWorld formula?
> >
> > pi<sup>(2)</sup>(x)=sigma(k=1,pi(sqrt(n)))
> [pi(n/p<sub>k</sub>-k+1]
>
> The MathWorld formula is expressed badly: I think it was hastily
> copiedwithout much understanding.
>
> > The following are a list questions I have:
> >
> > - pi<sup>(2)</pi>(x) is non-standard, at least unusual.
>
> I agree. It is also superfluous here: it is never referred
> to in the rest
> of the article. An editor would strike it.
>
> > Superscript usually means exponentiation.
>
> ...or functional iteration or.... The form here is most often
> seen with
> derivatives. Again, it is not doing anything useful.
>
> > - x should be n (I assume this is a typo)
>
> Yes
>
> > - Square brackets sometimes mean the nearest integer.
> > In this case I think it just denotes grouping; the author
> > could have used parens.
>
> Given the number theory context, I initially assumed [] meant nearest
> integer or floor. But the argument is integral, so the
> brackets are just
> used for grouping. (The pi function is (strangely) defined
> to have real
> arguments.) The real problem is the ill defined scope of the sum
> operator. The author could have avoided this by writing
>
> sum pi(n/p_k)-(k-1)
>
> Since both terms contain the index of summation, they are in the sum.
>
> Here is my rewrite of the formula (requires LaTeX):
>
> \documentclass{article}
> \begin{document}
> \thispagestyle{empty}\noindent
> Let $P$ be the set of primes $p$ such that $p^2<n$,
> and let $m=\left| P\right|$.
> Then the number of semiprimes less than or equal to $n$ is given by
> $$\sum_{p\in P} \left(\pi(n/p)-\pi(p)\right)=
> \sum_{p\in P} \pi(n/p)- \frac{m(m-1)}2.$$
> \end{document}
>
> > In J, with the same reasoning, one readily derives a computation
> > that produces the actual semiprimes. That is an illustration
> > of the power of J in dealing with arrays.
> >
>
> I agree.
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