Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $3n+1$ problem: an heuristic lower bound for the number of integers
  connected to 1 and less than $x$   This paper gives an heuristic lower bound for the number of integers
connected to 1 and less than $x$, $\theta(x) > 0.9x,$ in the context of the
$3n+1$ problem.

12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex