Table 2 Comparison of powerful number counting functions for \( N = 10^{m} \)
m | S (N) lower bound (14) | Q (N) | S (N)/q √N | Q (N)/q √N |
|---|---|---|---|---|
8 | 20,880 | 25,708 | 0.960773 | 1.182929 |
9 | 66,888 | 77,311 | 0.973282 | 1.124946 |
10 | 213,371 | 235,726 | 0.981806 | 1.08467 |
11 | 678,723 | 726,421 | 0.987604 | 1.057009 |
12 | 2,154,897 | 2,256,191 | 0.991555 | 1.038165 |
13 | 6,832,881 | 7,047,093 | 0.994247 | 1.025417 |
14 | 21,647,316 | 22,098,596 | 0.99608 | 1.016846 |
15 | 68,540,677 | 69,488,109 | 0.99733 | 1.011116 |
16 | 216,929,613 | 218,912,528 | 0.998181 | 1.007305 |
17 | 686,390,158 | 690,528,822 | 0.998761 | 1.004783 |
18 | 2,171,414,780 | 2,180,031,831 | 0.999156 | 1.003121 |
19 | 6,868,466,063 | 6,886,369,416 | 0.999425 | 1.00203 |
20 | 21,723,981,663 | 21,761,110,713 | 0.999608 | 1.001316 |
21 | 68,705,847,049 | 68,782,727,400 | 0.999733 | 1.000852 |
22 | 217,285,461,383 | 217,444,443,741 | 0.999818 | 1.00055 |
23 | 687,156,809,129 | 687,485,219,306 | 0.999876 | 1.000354 |
24 | 2,173,066,478,000 | 2,173,744,295,846 | 0.999916 | 1.000227 |
25 | 6,872,024,538,797 | 6,873,422,599,112 | 0.999942 | 1.000146 |