Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Combinatorial properties of sparsely totient numbers   Let $N_1(m)=\max\{n \colon \phi(n) \leq m\}$ and $N_1 = \{N_1(m) \colon m \in
\phi(\mathbb{N})\}$ where $\phi(n)$ denotes the Euler's totient function.
Masser and Shiu \cite{masser} call the elements of $N_1$ as `sparsely totient
numbers' and initiated the study of these numbers. In this article, we
establish several results for sparsely totient numbers. First, we show that a
squarefree integer divides all sufficiently large sparsely totient numbers and
a non-squarefree integer divides infinitely many sparsely totient numbers.
Next, we construct explicit infinite families of sparsely totient numbers and
describe their relationship with the distribution of consecutive primes. We
also study the sparseness of $N_1$ and prove that it is multiplicatively
piecewise syndetic but not additively piecewise syndetic. Finally, we
investigate arithmetic/geometric progressions and other additive and
multiplicative patterns like $\{x, y, x+y\}, \{x, y, xy\}, \{x+y, xy\}$ and
their generalizations in the sparsely totient numbers.

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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: econ