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Let $\mathrm{gnu}(n)$ denote the number of groups of order $n$ up to isomorphism.

$\mathrm{gnu}(1), \mathrm{gnu}(2), \mathrm{gnu}(3), \dots$ is now a sequence of integers, and we may ask if and where the subsequence $1, 2, 3, \dots, k$ appears successively, for some integer $k$.

Just looking through the first few terms, it is not hard to find an instance of $1, 2, 3, 4$, namely $\mathrm{gnu}(73), \mathrm{gnu}(74), \mathrm{gnu}(75), \mathrm{gnu}(76)$. But $\mathrm{gnu}(77)=1$, so this is not a $1,2,3,4,5$.

To find $1,2,3,4,5$ is significantly harder. There are necessary and sufficient conditions on $n$ which determine whether $\mathrm{gnu}(n)=1,2,3,4$, listed for example in this paper on page 6. I implemented these to find the integers $n$ such that $\mathrm{gnu}(n+i)=i$ for $1 \leq i \leq 4$. I've written a computer program to find these - the sequence begins $n = 72, 20664, 66600, 84744, 89784, 141240, 175032, 232680, 271272, 288072, 378984,...$.

Hence our candidates for where we want to find a $5$ are at $77, 20669, 66605, 84749, 89789, 141245, 175037, 232685, 271277, 288077, 378989,...$.

I've tested these manually and, hoping that my calculations and program are correct, the earliest instance of a $1,2,3,4,5$ occurs at $\mathrm{gnu}(2814121), \mathrm{gnu}(2814122), \mathrm{gnu}(2814123), \mathrm{gnu}(2814124), \mathrm{gnu}(2814125)$. But $\mathrm{gnu}(2814126)=24$, so this is not a $1,2,3,4,5,6$.


The obvious next challenge is to find a $1,2,3,4,5,6$. That's a lot harder as:

  • I've struggled to find a necessary and sufficient condition for $\mathrm{gnu}(n)=5$ anywhere online (and am not sure on how to begin deriving it myself!)
  • As $n$ gets large, the tests for whether $\mathrm{gnu}(n)=1,2,3,4$ tend to get longer, since they're based on the prime factorization of $n$

So, I'm wondering if anyone has any observations that'd make finding a $1,2,3,4,5,6$ easier:

  • Is there a necessary and sufficient condition for $\mathrm{gnu}(n)=5$?
  • Is there some smarter method for how we may search for this?
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    $\begingroup$ I have to admit that I cannot really understand why lots of people find this question (and also a related question,"does there exits $n>1$ with ${\rm gnu}(n)=n$?") interesting. Is it just curiosity? $\endgroup$
    – Derek Holt
    Commented Jun 12 at 17:44
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    $\begingroup$ @DerekHolt There are tons of boring questions and overmore heavily opinion-based , and then there occurs a really interesting question like this (namely whether the sequence of gnus contains $1,2,3,4,5,6$ and , if , when it occurs first) and you find this "boring" ? Seems that people have a very different opinion concerning "interesting" and "boring". The upvotes need not mean that the users consider this "interesting" , the excellent presentation alone deserves a high score. We need more of such questions ! $\endgroup$
    – Peter
    Commented Jun 13 at 14:24
  • $\begingroup$ The related question was posted by me but I was not the first. It is an old still open problem whether $gnu(n)=n$ has a nontrivial solution , but I did not know this. I independently arrived at this question which I consider both natural and fascinating. $\endgroup$
    – Peter
    Commented Jun 13 at 14:30
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    $\begingroup$ @Peter I did not use the word "boring", and I do not find the problem boring. I was hoping that someone would say why they find the question interesting. $\endgroup$
    – Derek Holt
    Commented Jun 13 at 16:07
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    $\begingroup$ Similarly to many elementary questions in number theory (for example: are there infinitely many primes of the form $n^2+1$), it can be very difficult to say anything interesting about questions of this type. The question might become interesting if a solution is found.. $\endgroup$ Commented Jun 15 at 10:25

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