0

It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $n$ is $$\pi(n) \sim \int_2^n \frac{dt}{\log(t)}.$$

Question: Are there other sequences that have a distribution such that the likelihood of $n$ being a member of that sequence is approx $1/\log(n)$?

Further Question: What properties or constraints would such sequences need to adhere to?

Gary
  • 31,845
Penelope
  • 3,147
  • Thanks everyone for the replies. The use of the "floor function" is certainly valid, but I wonder if there are sequences that don't make use of the floor function, but are constructed using only the simple operations (multiplication, addition) eg $x_n=n^2$, or using simple rules like the fibonnaci sequence $x_n=x_{n-1}+x_{n-2}$ – Penelope Feb 28 '24 at 15:44

3 Answers3

2

One such sequence is $a_n = \lfloor n\log n\rfloor$. (In a similar way, one can construct sequences with any desired density.)

Another is $\{n\in\Bbb N\colon n$ is divisible by $\lfloor \log n\rfloor\}$ (which also has obvious generalizations).

Greg Martin
  • 78,820
2

You can construct a lot of sequences from the primes themselves that have this density. For instance, if you exclude all primes of the form $n^2+1$, since the density of the perfect squares decreases faster than that of the primes, you still asymptotically have density $\frac1{\log n}$. (In fact, in this case it’s not known whether infinitely many primes are excluded; see Primes of the form $n^2+1$ - hard?)

Or, perhaps slightly more interesting: The density of $k$-almost primes (i.e. integers with $k$ prime factors) is asymptotic to

$$ \frac{(\log\log n)^{k-1}}{(k-1)!\log n} $$

(see asymptotic density of $k$-almost primes). You can combine this with Greg Martin’s approach: The sequence you get from the sequence of $k$-almost primes by omitting $\left\lfloor\frac{(\log\log n)^{k-1}}{(k-1)!}\right\rfloor$ terms after a term $n$ has density $\frac1{\log n}$.

joriki
  • 238,052
0

There is a lot of such sequences. For example:

$p_n +1$

$p_n +7$

$p_n + (-1)^n$

jjagmath
  • 18,214