TITLE: What is the meaning of the symbol "$\left|\underline{\;n\;}\right.$" for integer $n$?
QUESTION [39 upvotes]: For example,

(6) The sequence of primes is endless.
For, if $p$ is any prime, the number ${\begin{array}{|c}\color{red}p\\\hline\end{array} + 1}$ is greater than $p$ and is not divisible by $p$ or by any smaller prime. If then ${\begin{array}{|c}p\\\hline\end{array} + 1}$ is not a prime, it must have a prime divisor greater than $p$, and in either case a prime greater than $p$ exists.

And here is another one:

9. Theorems.
(1) The product of any $n$ consecutive integers is divisible by $\begin{array}{|c}n\\\hline\end{array}$
For $(m+1)(m+2)...(m+n)/\begin{array}{|c}n\\\hline\end{array} = \begin{array}{|c}\color{red}{m + n}\\\hline\end{array}/\begin{array}{|c}\color{red}m\\\hline\end{array}\begin{array}{|c}\color{red}n\\\hline\end{array}$, and to show that the last expression is an integer it is sufficient to show that any prime $p$ which occurs in $\begin{array}{|c}\color{red}m\\\hline\end{array}\begin{array}{|c}\color{red}n\\\hline\end{array}$ occurs to at least as high a power in $\begin{array}{|c}\color{red}{m+n}\\\hline\end{array}$. Thus we have to show that
$$I[(m+n)/p]+I[(m+n)/p^2]+I[(m+n)/p^3]+... \\
\geq I[m/p]+I[m/p^2]+I[m/p^3]+... \\
+I[n/p] +I[n/p^2]+I[n/p^3]+...$$

REPLY [16 votes]: Adding to @Bernard Masse, the history of notations for products of terms in arithmetical progression is quite long. The following is taken from Christian Kramp, 1808, Elémens d'arithmétique universelle (pdf), p. XI-XII:


Pour désigner les produits dont les facteurs constituent entr’eux une progression arithmétique, tels que $a(a+r)(a+2r)....(a+nr-r)$, j’ai conservé la notation $a^{n|r}$, déjà proposée dans mon analyse des refractions; je leur avois donné le nom de facultés. Arbogast lui avoit substitué la dénomination plus nette et plus françoise de factorielles; j’ai reconnu l’avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage à la mémoire de mon ami.

where  $1^{p|1}$ would denote the (nowadays) factorial.
He used the term "faculties (facultées en French)", acknowledged Louis Arbogast for the term "factorial (factorielles)", and on page 348, he introduces the notation $p!$ for $1^{p|1}$ (additional details in History of notation: "!"). You can find this from Article 445 (Volume 2, Page 66) in Florian Cajori, 1993, A History of Mathematical Notations (Dover Publications).
From Article 447, Cajori mentions Thomas Jarrett (1805–1882) for his extensive study of algebraic notations, An essay on algebraic development: containing the principal expansions in common algebra, in the differential and integral calculus, and in the calculus of finite differences, 1831, Page 15,  Article 38, you find:
$$\begin{array}{|c}p\\\hline\end{array}=p(p-1)\ldots 1$$


The symbol $\underset{n,m}{\begin{array}{|c}a\\\hline\end{array}}$ denotes the product of $n$ factors forming an arithmetical progression, of which the first term is $a$, and the common difference $m$; if $m = -1$, the $m$ may be omitted; and if, in the same case, $n = a$ the $n$ also may be omitted: thus
$\underset{n,m}{\begin{array}{|c}a\\\hline\end{array}} = a(a+m)(a+2m)...(a+\overline{n-1}.m)$,
$\underset{n\phantom{,m}}{\begin{array}{|c}a\\\hline\end{array}} = a(a-1)(a-2)...(a-n+1)$, and
${\begin{array}{|c}a\\\hline\end{array}} = a(a-1)(a-2)...2.1$.