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I've been thinking about primorials in the context of the twin prime conjecture. I am seeing this primarily as an exercise to improve my intuition about primorials and prime patterns more than the possibility of making any progress. :-)

So, I thought that it would be interesting to think about numbers of the following form. Let $p_k$ be the $k$th prime. Then, I am interested in the following sequence of numbers:

$\left(p_k\#-1, p_k\#+1\right)$, $\left(\frac{p_k\#}{3}+1,\frac{p_k\#}{3}+3\right)$, $\left(\frac{p_k\#}{3}-1,\frac{p_k\#}{3}-3\right)$, $\left(\frac{p_k\#}{2}+2,\frac{p_k\#}{2}+4\right)$, $\left(\frac{p_k\#}{2}-2,\frac{p_k\#}{2}-4\right)$, $\left(\frac{p_k\#}{15}+3,\frac{p_k\#}{15}+5\right)$, $\left(\frac{p_k\#}{15}-3,\frac{p_k\#}{15}-5\right) \cdots$

Here is my question. Does anyone have any suggestions on methods for counting the number of distinct integeres in this sequence? My basic goal here is to estimate a lower bound for the number of distinct $x$ where $p_k< x \le p_k\#+1$

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I'm not clear on the definition of the sequence, could you be more specific? –  Charles Jul 25 '13 at 14:56
The sequence is an effort is to use the primorial to create a sequence of possible twin prime pairs. So the first value is (-1,+1), the next one is (+1,+3) and (-1,-3), the one after that is (+2,+4) and (-2,-4) etc. I hope that explains the sequence. For my purposes, I would be fine if you provide a lower estimate of just the positive values of the sequence. –  Larry Freeman Jul 26 '13 at 2:40
You're still not explaining where -1, +1, +1, +3, -1, -3, +2, +4, -2, -4, etc. come from. –  Charles Jul 26 '13 at 13:46
Thanks for your question. The sequence is just counting. I am going 1,2,3,4,5,6 ,etc. then I am adding the sequence number to a primorial divided by the common factors between the sequence number and the primorial. So, assume we have a form of $\frac{p_k}{a} + c,\frac{p_k}{a}+c+2$ where $a | p_k\#$ and $a | c(c+2)$. Does that answer explain where I am going? –  Larry Freeman Jul 26 '13 at 22:30

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up vote 1 down vote accepted

You asked for a bound on the number of integers between $p_k$ and $p_k\#+1$. $\log x\#\sim x$ and so the number is roughly $e^{p_k}\approx k^k.$

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Wow. Thanks very much for the answer! I would be very interested if you could provide details explaining your results. –  Larry Freeman Jul 26 '13 at 2:38
To be clear, I really like your analysis for showing that $p_k$# is approximately $k^k$. I am interested in understanding how this applies the number of distinct integers in the sequence that I describe. Are you saying the lower bound of the sequence is $k^k$. If so, I would be very interested in understanding your reasoning. –  Larry Freeman Jul 26 '13 at 3:50
It's approximate, not a lower bound. If you replace e with any smaller number it becomes a lower bound for all but finitely many numbers. The prime number theorem says that x# is about exp(x) and p_k is about k log k; combining the two gives the result. –  Charles Jul 26 '13 at 13:45

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