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Heuristically speaking, there should be infinitely many such primes, by the same sort of heuristic as one would use to expect there to be infinitely many Mersenne primes. That is, by the Prime Number …

This question might be difficult, but heuristically it should be true. Essentially the same heuristic that yields the Artin primitive root conjecture, there should be for any such $a$ and $b$ be infin …

Based on standard heuristics, one should expect there to be infinitely many such $n$ (Edit: Actually see below). By the prime number theorem, the "chance" that $2^{2n}-2^n+1$ should be prime should be …

Heuristically, one would expect the answer to be yes. There's an existing partially explicit version of this mentioned in Richard Guy's "Unsolved Problem in Number Theory" entry F24, which is that for …

Heuristically, we should expect only finitely many such numbers. If $C(x)$ is the number of highly composite numbers which are less than or equal to $x$, then Erdos and Nicholas showed that there are …

Since the term multiply perfect is as you acknowledge already in the literature, I'm not sure why you are introducing the baroque term. One noteworthy thing here is what are called pseudoperfect numbe …

See Chen's 1975 article "On the Distribution of Almost Primes in an Interval." This proves Legendre's conjecture for almost primes (primes or semiprimes) and for sufficiently large $n$. See also a tig …

Recall, a collection of congruences with distinct moduli, each greater than 1, such that each integer satisfies at least one of the congruences, is said to be a covering system (or a covering system o …

Not a complete answer, but here is a proof that every one of them of the form $2^k p^a$ where $p$ is an odd prime is of the form in question. Note that all even perfect numbers are of the form $2^{q-1 …

Recall a set of integers $S$ is said to be an additive basis for the natural numbers if there is a $k$ such that every positive integer is expressible as a sum of at most $k$ elements of $S$. Similarl …

Given a real number $r$, and an integer $b$>0, we can define $B_b(r)$ as the set of numbers which are obtained from $r$ by writing $r$ in base $b$ and then altering a density zero subset of its digits …

Given a fixed $a$ and $q$, this should be true for sufficiently large $n$ by the explicit versions of Dirichlet's theorem on arithmetic progressions. We cannot prove that this is true for every $n$. T …

Yes, those are the only solutions. To see this note that $$1+3+9+27 \cdots 3^m = \frac{3^{m+1}-1}{2}.$$ So we are looking for solutions of $\frac{3^{m+1}-1}{2}=2^n$, or equivalently looking for soluti …

The set $S$ is very likely finite. It is unclear if you intend for $a$, $b$, $c$ and $d$ to be positive. If you don't assume that $a$, $b$, $c$ and $d$ are positive, then $n!$ has such a representati …

As Wojowu notes in their comment, given the state of the art, we can't even prove that there are infinitely many primes $p$ where $M_p$ is composite. That said, standard heuristic arguments support th …