5
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another sequence on twin primes The maximal prime factor of average of twin prime pair:

n = 100000; averageList = Select[Prime[Range[n]], PrimeQ[# + 2] &] + 1; mpfList = FactorInteger[#][[-1, 1]] & /@ averageList; ListPlot[%]

Why does this diagram look like a series of trajectories? alt text

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    Never mind about the plot, if my answer below is sufficient.2010-12-11

1 Answers 1

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The nth twin prime pair appears to be around $c n \log^2 n$ for a constant $c$, large $n$. The largest prime factor of any non-prime $m$ is equal to $n/m'$ where $m'\ge2$. What you see in the plot are approximations of the curves $y=\frac{c}{k} x \log^2 x$ for $k=6,12,24,...$, since the mean of a twin pair is a multiple of 6.

Added: Notice the same effect when plotting simply the maximum prime factor of 6n versus n: alt text

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    That should be $f(n)=\frac{cn \log^2 n}{\log \log \log n}$2010-12-12