OEIS conjecture that the Hofstadter consecutive-sum sequence omits infinitely many integers

Let $(a_n)_{n\ge1}$ be the strictly increasing sequence defined by

$$a_1=1,\qquad a_2=2,$$

and, for $k\ge3$, let $a_k$ be the least integer greater than $a_{k-1}$ that can be written as a sum of at least two consecutive earlier terms:

$$a_k=\sum_{i=p}^{q}a_i$$

for some $1\le p\le q\le k-1$ with $q-p\ge1$. OEIS conjecture. The sequence $(a_n)_{n\ge1}$ omits infinitely many positive integers. The conjecture concerns the distribution of the classical Hofstadter consecutive-sum sequence, whose asymptotic behavior was previously posed as an open problem. The paper proves this assertion, so the conjecture is now settled.