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Consider $X=\{q(q^n-1)|q \ \text{is some power of a prime number}, n\in \Bbb N^*\}$, $S=\{ s \in \Bbb N| s=\prod_i s_i, s_i \in X\} $, I am interested in which integers are in $S$. For example, $2, 6 \in S$ while $5 \notin S$.

  1. Does $S$ have a positive density?
  2. Does there exist arbitrary long consecutive sequences in $S$?
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  • $\begingroup$ Uh, why is 2 in S and 4 not in S? Gerhard "S Is Not Multiplicatively Closed?" Paseman, 2018.06.12. $\endgroup$
    – Gerhard Paseman
    Commented Jun 12, 2018 at 19:02
  • $\begingroup$ Are you requiring the $s_i$ to be distinct? $\endgroup$
    – Robert Israel
    Commented Jun 12, 2018 at 21:06
  • $\begingroup$ Sorry, I have edited it to make it clear that $s_i$ can be the same. $\endgroup$
    – Zhiyu
    Commented Jun 12, 2018 at 22:25
  • $\begingroup$ All members of $X$ are even, so all members of $S$ are even (unless you count $1$ as the empty product), so you can't have even two consecutive integers in $S$ (except $1,2$ if you count $1$ as the empty product). $\endgroup$
    – Robert Israel
    Commented Jun 13, 2018 at 20:40

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By my calculations, there are $1254$ members of $S$ up to $10^5$. Plots seem to indicate $S_n$ growing faster than linearly with $n$, perhaps like $c n^2$ for some constant $c$. So the asymptotic density appears to be $0$.

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  • $\begingroup$ Could you please delete this answer? I found the problem boring so I want to delete it. $\endgroup$
    – Zhiyu
    Commented Jun 24, 2018 at 7:22
  • $\begingroup$ Please don't delete it. When somebody has taken the time and trouble to answer a question, it's very annoying to have it disappear. $\endgroup$
    – Robert Israel
    Commented Jun 24, 2018 at 19:39
  • $\begingroup$ Fine, I agree with you. $\endgroup$
    – Zhiyu
    Commented Jun 26, 2018 at 15:51

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