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Statement

Here $f(n)$ is the number of representations of $n$ as a sum of one or more consecutive primes, whose average is $\log2$ by equation (5) (p. 160), so that $f(n)=0$ for infinitely many $n$ (p. 161).

Problems (p. 161). The paper lists the following among the problems that suggest themselves, numbered as printed:

1. Is $f(n)=1$ for infinitely many $n$?
2. Is $f(n)=k$ solvable for every $k$?
3. For every $k$, does the set of $n$ with $f(n)=k$ have a density?
4. Is $\limsup_{n\to\infty}f(n)=\infty$? (The print writes the upper limit as $\overline{\lim}\,f(n)=\infty$.)

The paper answers none of them.

Source. L. Moser, Notes on Number Theory III: On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), no. 2, 159--161, DOI 10.4153/CMB-1963-013-1, p. 161. The edition read is identified on the source card.

Read depth. Claims checked: the four problems were read on the printed page.

Proof pointer

None: open problems as posed.

Dependencies

Equation (5), which supplies the fact that $f(n)=0$ infinitely often.

Bears on

• Problem 358: the problems concern the problem's $f(n)$ for the single sequence of primes, which by equation (5) has $f(n)=0$ infinitely often and so answers neither of Problem 358's questions. Problem 4 asks only whether $f(n)$ is unbounded for the primes, a weaker property than the $f(n)\to\infty$ that Problem 358 asks of some sequence. The paper does not mention Erdős or the problem.