# Does each integer $n>1$ have the form $2^k3^l+p_m-p_{m-1}+\ldots+(-1)^{m-1}p_1$?

For each positive integer $$n$$, let $$s_n$$ be the alternating sum of the first $$n$$ primes given by $$s_n:=p_n-p_{n-1}+\ldots+(-1)^{n-1}p_1,$$ where $$p_k$$ denotes the $$k$$th prime. All the numbers $$s_1,s_2,\ldots$$ are pairwise distinct. The main term of $$s_n$$ as $$n\to\infty$$ is not known. It seems that $$\lim_{n\to\infty}s_n/p_n=1/2$$.

Here I ask a novel question involving $$s_n$$ on the basis of my computation.

Question. Is it true that $$\{2^k3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,3,\ldots\}=\{2,3,\ldots\}?$$

I conjecture that the question has a positive answer. Let $$r(n)$$ be the number of ways to write $$n$$ as $$2^k3^l+s_m$$ with $$k,l\in\{0,1,\ldots\}$$ and $$m\in\{1,2,\ldots\}$$. The sequence $$r(1),r(2),\ldots$$ is available from http://oeis.org/A308411. For example, $$r(2)=1$$ with $$2=2^03^0+p_2-p_1$$. On May 25, 2019 I verified $$r(n)>0$$ for all $$n=2,\ldots,10^6$$. On May 26, 2019 Prof. Qing-Hu Hou extended the verification to $$2\times 10^7$$ on my request. Based on Hou’s program I have verified that $$r(n)>0$$ for all $$n=2,\ldots,10^9$$.

PS: By the way, I also note that the set $$\{6^k+3^l+s_m:\ k,l=0,1,\ldots\ \text{and}\ m=1,2,\ldots\}$$ contains $$3,4,\ldots,10^9$$, and conjecture that this set coincides with $$\{3,4,\ldots\}$$ (cf. http://oeis.org/A308403).