My question is: is it common to say "I can't seem to do it" when one
really means "It seems I can't do it", and is it "correct" (in
whatever sense that makes sense)?
Yes, it's common, and the difference is that the first expresses a certain level of frustration, as in, "I've looked everywhere and I can't seem to find it."
The second construction would express a certain level of resignation, "It seems I can't find it, so we'll have to do without it."
The two can also be used interchangeably when the level of either frustration or resignation is low.
Both expressions are very common and universally understood in mathematical English written by native speakers -- which means that that, as a practical matter they are both right. And if a dictionary claims that "denote" can't be used in this way, that is the dictionary's fault.
Your assumption that (2) requires that the symbol has already been mentioned does not match how these phrasings are used in practice. Indeed, a sentence such as (2) would commonly be found in an early section of a paper entitled "mathematical preliminaries" or something like that, where the author briefly summarizes the symbolism that he's going to use in the rest of the paper.
As a writer your wording would probably sound most natural if you use the bare statement of fact "N denotes ..." for notation that already exists, such that the sentence just reminds the reader what is already true, whereas "We denote ..." or "Let N mean ..." and so forth are for definitions that are part of the original content of what you're writing, and therefore are new to the reader.
However, as a reader you shouldn't try to extract any meaning at all from the difference between these phrasings. The trend I just sketched here is not followed consistently enough for that to work. Go by context instead.
By the way, your third sentence
We let N denote the set of natural numbers.
sounds slightly clunky to me; simply
Let N denote the set of natural numbers.
would be more straightforward.
(Note that actually neither of your sentences could usefully be used in a mathematical paper, since there is no real doubt that N and "the set of natural numbers" means the same, whereas differing traditions about whether zero counts as a natural number or not. Defining N to mean the set of natural numbers without revealing whether this set includes 0 for you is just a waste of ink and/or taunting the reader).
Best Answer
First off, I think that the verb to count doesn't work well in the context of making mathematicalензщ calculations using your intellectual ability, in other words--solving math problems mentally. The process may also be reffered to as "mental maths" or mental arithmetic that is doing calculations in your mind, without writing down any numbers. If you google for "calculate mentally" or "mentally calculate", you'll find lots of examples.
Also, see the " in one's head" phrase which in your case hits the bull's eye, as Tromano pointed out in his comment: