Multiply gleeful numbers

Abstract: For positive integers $k$ and $n$ let $f_k(n)$ be the number of ways of representing $n$ as a sum of $k$ powers of consecutive primes. A number is called $k$-gleeful if $f_k(n)>0$ and multiply gleeful if $f_k(n)>1$ or $f_k(n)f_{k'}(n)>0$ for some positive integers $k< k'.$ Under Schinzel's hypothesis H, we show that there are infinitely many positive integers $n$ such that $f_2(n)f_4(n)>0$. Under the same assumption we show that $\limsup_{n\to\infty} f_2(n)=\infty$. This gives a conditional proof of a stronger version of a conjecture of Moore and Sorenson from the preprint arXiv:2507.09012v1, July 2025.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)