Abstract
AbstractLet M(x) denote the largest cardinality of a subset of $$\{n \in \mathbb {N}: n \le x\}$$
{
n
∈
N
:
n
≤
x
}
on which the Euler totient function$$\varphi (n)$$
φ
(
n
)
is nondecreasing. We show that $$M(x) = \left( 1+O\left( \frac{(\log \log x)^5}{\log x}\right) \right) \pi (x)$$
M
(
x
)
=
1
+
O
(
log
log
x
)
5
log
x
π
(
x
)
for all $$x \ge 10$$
x
≥
10
, answering questions of Erdős and Pollack–Pomerance–Treviño. A similar result is also obtained for the sum of divisors function $$\sigma (n)$$
σ
(
n
)
.
Funder
Directorate for Mathematical and Physical Sciences
Publisher
Springer Science and Business Media LLC
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