We study expressions of the type <!-- MATH
 $\prod_{p}
p^{\lfloor\frac{x}{f(p)}\rfloor}$
 -->
<SPAN CLASS="MATH"><IMG STYLE="height: 3.47ex; vertical-align: -1.08ex; " SRC="img2.svg"
 ALT="$\prod_{p}
p^{\lfloor\frac{x}{f(p)}\rfloor}$"></SPAN>, where <SPAN CLASS="MATH"><IMG STYLE="height: 1.19ex; vertical-align: -0.10ex; " SRC="img3.svg"
 ALT="$x$"></SPAN> is a nonnegative real number,
<SPAN CLASS="MATH"><IMG STYLE="height: 2.25ex; vertical-align: -0.55ex; " SRC="img4.svg"
 ALT="$f$"></SPAN> is an arithmetic function satisfying some conditions, and the product
is over the primes <SPAN CLASS="MATH"><IMG STYLE="height: 1.64ex; vertical-align: -0.55ex; " SRC="img5.svg"
 ALT="$p$"></SPAN>. We begin by proving that such expressions can be
expressed by using the <!-- MATH
 $\mathrm{lcm}$
 -->
<SPAN CLASS="MATH"><IMG STYLE="height: 1.80ex; vertical-align: -0.10ex; " SRC="img6.svg"
 ALT="$\mathrm{lcm}$"></SPAN> function, without reference to
prime numbers; we illustrate this result with several examples. The rest
of the paper is devoted to studying two particular cases related
to <SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="img7.svg"
 ALT="$f(m) = m$"></SPAN> and <!-- MATH
 $f(m) = m - 1$
 -->
<SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.68ex; " SRC="img8.svg"
 ALT="$f(m) = m - 1$"></SPAN>. In both cases, we find arithmetic
properties and analytic estimates for the underlying expressions.