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Suppose the divisors of $n$ greater than $1$ begin $d_1<\cdots<d_k$, with sum $N$. Put $L=\operatorname{lcm}(d_1,\ldots,d_k)$. Then $L\mid n$, so $L\le n$. Every $d_i$ divides $L$. Conversely, if $2\le e\le d_k$ divides $L$, it divides $n$ and hence is one of the $d_i$, since these are the entire prefix through $d_k$. Thus $L$ has precisely the same divisor prefix.

Every original witness therefore gives a partition $P$ of $N$ into distinct integers at least $2$ such that the divisors of $\operatorname{lcm}(P)$ up to $\max(P)$ are exactly $P$, and gives such a witness of size no larger than the original integer. Conversely, any partition passing this test realizes its sum as a divisor prefix of $\operatorname{lcm}(P)$. Taking minima proves the formula, including the convention that an empty family means no witness exists. Every part is at most $N$, so each candidate least common multiple divides $\operatorname{lcm}(1,\ldots,N)$; there are finitely many partitions.

The checker enumerates all increasing distinct-part partitions of each $N$ in $[2,64]$. It verifies partition validity, uniqueness and counts against a separate subset-sum dynamic program, tests the exact divisor-prefix predicate, and minimizes the candidate least common multiples. The retained table contains every minimum and one corresponding prefix. A separate divisor sieve for $n\le10000$ crosschecks all the attained minima; completeness over all $n$ comes from the reduction, not from this sieve bound.

There are 83,691 tested partitions. Only $N=4$ has no valid prefix. The largest minimum is $f(58)=265$, with prefix $(5,53)$, whose least common multiple is $265$. The boundary values are $f(2)=2$, $f(3)=3$, $f(5)=5$ and $f(6)=4$. These exact finite conclusions are the whole computational scope.