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https://en.wikipedia.org/wiki/English_articles#Indefinite_article

The indefinite article of English takes the two forms a and an. Semantically, they can be regarded as meaning "one", usually without emphasis. They can be used only with singular countable nouns; for the possible use of some (or any) as an equivalent with plural and uncountable nouns, see Use of some below.

It says if a noun comes with an indefinite article, the noun is singular and countable.

Then there are phrases such as 'a lot of', 'a number of', and 'once upon a time'. Are 'lot', 'number', 'time' above countable nouns as well? I have never heard of people saying 'two numbers of people' or 'once upon three times', which makes me doubt that they are actually countable.

Thunderweb
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    If I’ve told you once, I’ve told you a thousand times: if you can count it, it’s countable. Like the time we ordered six lots of pickled weasels — wait, we did that twice — so the two times we ordered six lots of pickled weasels. But the vendor sure did pull a number on us, remember, because the weasels weren’t pickled, merely dill. In fact he pulled two numbers on us because the weasels weren’t even weasels but stoats. But they did those little dance numbers, you know? With the hats? Anyway it all worked out in the end, and we learned we can count on each other. Multiple times. – Dan Bron Dec 10 '17 at 12:17
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    In almost all cases, "a" selects a count noun, but under restricted conditions it can combine with a non-count singular: "a number of problems were found"; "Ed has a good knowledge of Latin"; " He wastes a good deal of time". – BillJ Dec 10 '17 at 12:34
  • @BillJ So that 'number', 'knowledge', and 'deal' in your examples are uncountable nouns, even if they are combined with 'a'? – Thunderweb Dec 10 '17 at 12:49
  • Yes, as quantificational nouns "number" and "deal" are non-count. And "knowledge" has no established plural. – BillJ Dec 10 '17 at 12:53
  • 'The director spoke at the meeting today with five enormous enthusiasms' is unacceptable whereas both 'The director spoke at the meeting today with enormous enthusiasm' and 'The director spoke at the meeting today with an enormous enthusiasm' are grammatical. – Edwin Ashworth Dec 10 '17 at 16:12
  • The numbers don't lie. "Lots" and "numbers" are both countable things. Even at several times demonstrates countable "times". – Gary Botnovcan Dec 10 '17 at 16:58
  • @Gary Botnovcan Please read BillJ's second comment. As [a] quantificational noun[ ], 'number' is non-count. // An easier example to understand is 'Coffee is my favourite drink' (a non-count usage) and 'The two major coffees are arabica and robusta' (a count usage). – Edwin Ashworth Dec 10 '17 at 17:29
  • @EdwinAshworth In "a number of men" - "number" in my opinion is a count noun. "Number" is telling us that it was one of 3,4,5,6, etc. It wasn't all of those numbers. It was just one of them - "a number of men". We just don't know which number it was, but it was "a number". – WS2 Dec 11 '17 at 22:16
  • You're not using the CGEL interpretation of countness. Or any other one I've ever come across. You don't use 'Two numbers of men were present at the weekend.' any more than you use 'I have two piles of jobs to do.' As a quantificational noun, 'number' is here part of the compound quantifier 'a number of' which is paraphrasable by simple quantifiers like some / several. – Edwin Ashworth Dec 19 '17 at 11:57

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In comments, BillJ wrote:

In almost all cases, "a" selects a singular count noun, but under restricted conditions it can combine with a non-count singular: "a number of problems were found"; "Ed has a good knowledge of Latin"; " He wastes a good deal of time".

Yes, as quantificational nouns "number" and "deal" are non-count. And "knowledge" has no established plural.

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