Is the group-enumeration (gnu) function surjective onto the positive integers? (Kourovka 15.99)

Let $f(n)$ denote the number of isomorphism classes of finite groups of order $n$ — the 'gnu' (group number) function. Is it true that for every positive integer $k$ the equation $f(n)=k$ has a solution — that is, is $f$ surjective onto $\mathbb{Z}_{>0}$? Equivalently, is every positive integer the number of groups of some order? This is known to hold for all $k\le 1000$ (Wei, 1998). This is W. J. Shi's group-enumeration surjectivity question; it is unrelated to the separately-proved Shi recognition conjecture that a finite simple group is determined by its order together with its set of element orders.

Acceptance. FULLY RESOLVES: EITHER a proof that f is surjective — for instance a uniform construction that, given any k, produces a squarefree n with f(n)=k using Hölder's formula together with Dirichlet's theorem on primes in arithmetic progressions — OR a proof that some specific positive integer k is never the number of groups of any order. ADVANCES: a certified extension of the verified range beyond k=1000 — a re-runnable computation exhibiting, for every k <= K with K > 1000, an explicit witness order n with f(n)=k (Hölder's formula on squarefree n suffices, and f(n) is then machine-checkable), plus the search code; the larger K, the stronger the advance. Isolating the structural obstruction to a uniform construction (e.g. which residue-class patterns of primes realize a prescribed k) is a partial advance.