Note: As it is not possible to find k space tables online (everything is based on log scale counts of primes and twin primes in n space), I built out some tables up to k=10^15 (which is equivalent to n=6(10^15)+1 in n space) using existing literature and Primesieve.
It would not have been practical to do this with even the optimized segmented k space sieve I built for the k=10^12 run. Primesieve is a highly optimized C++ sieve. It is way faster than anything I built, but all it does is count in n space. So while the vanilla version isn’t useful for visualizations, or computing the Hardy Littlewood constants directly from raw data, Primesieve is the superior tool for counting.
I could probably run these using Primesieve for larger numbers as well, but it could take months potentially, and I’d have to leave my computer on. It took around 3 days on a 10 core M4 Mac to find the n=6(10^15)+1 counts (which I will note, was also — surprisingly for me anyway — way faster than Primesieve running on the 28-thread Intel i7-14700f I did the prior empirical work with).
The first few values, 0,1,2,3 are also included in order to show the different behavior of counts of primes and twin prime yielding k indices across the whole scale.
| n range 0 to ? | No of Primes | % of n Prime |
| 0 | 0 | 0.00000000% |
| 1 | 0 | 0.00000000% |
| 2 | 1 | 50.00000000% |
| 3 | 2 | 66.00000000% |
| 10 | 4 | 40.00000000% |
| 100 | 25 | 25.00000000% |
| 1,000 | 168 | 16.80000000% |
| 10,000 | 1,229 | 12.29000000% |
| 100,000 | 9,592 | 9.59200000% |
| 1,000,000 | 78,498 | 7.84980000% |
| 10,000,000 | 664,579 | 6.64579000% |
| 100,000,000 | 5,761,455 | 5.76145500% |
| 1,000,000,000 | 50,847,534 | 5.08475340% |
| 10,000,000,000 | 455,052,511 | 4.55052511% |
| 100,000,000,000 | 4,118,054,813 | 4.11805481% |
| 1,000,000,000,000 | 37,607,912,018 | 3.76079120% |
| 10,000,000,000,000 | 346,065,536,839 | 3.46065537% |
| 100,000,000,000,000 | 3,204,941,750,802 | 3.20494175% |
| 1,000,000,000,000,000 | 29,844,570,422,669 | 2.98445704% |
| 10,000,000,000,000,000 | 279,238,341,033,925 | 2.79238341% |
| 100,000,000,000,000,000 | 2,623,557,157,654,233 | 2.62355716% |
| 1,000,000,000,000,000,000 | 24,739,954,287,740,860 | 2.47399543% |
| k range 0 to ? | No of Twin Prime Pairs (n=6k-1, n+2= 6k+1) | % of k makes a twin prime in n=6k+-1 |
| 0 | 0 | 0.00000000% |
| 1 | 1 | 100.00000000% |
| 2 | 2 | 100.00000000% |
| 3 | 3 | 100.00000000% |
| 10 | 6 | 60.00000000% |
| 100 | 26 | 26.00000000% |
| 1,000 | 142 | 14.20000000% |
| 10,000 | 810 | 8.10000000% |
| 100,000 | 5,330 | 5.33000000% |
| 1,000,000 | 37,915 | 3.79150000% |
| 10,000,000 | 280,557 | 2.80557000% |
| 100,000,000 | 2,166,300 | 2.16630000% |
| 1,000,000,000 | 17,244,408 | 1.72444080% |
| 10,000,000,000 | 140,494,396 | 1.40494396% |
| 100,000,000,000 | 1,166,916,932 | 1.16691693% |
| 1,000,000,000,000 | 9,846,842,483 | 0.98468425% |
| 10,000,000,000,000 | 84,209,699,420 | 0.84209699% |
| 100,000,000,000,000 | 728,412,916,122 | 0.72841292% |
| 1,000,000,000,000,000 | 6,363,082,106,476 | 0.63630821% |
| k Range: k>=10 | n Range: (k*6)+1 | No of Primes Less than or Equal to n in Range | No of Twin Prime Pairs (6k-1,6k+1) Less than or Equal to n in Range |
| 10 | 61 | 18 | 6 |
| 100 | 601 | 110 | 26 |
| 1,000 | 6,001 | 783 | 142 |
| 10,000 | 60,001 | 6,057 | 810 |
| 100,000 | 600,001 | 49,098 | 5,330 |
| 1,000,000 | 6,000,001 | 412,849 | 37,915 |
| 10,000,000 | 60,000,001 | 3,562,115 | 280,557 |
| 100,000,000 | 600,000,001 | 31,324,704 | 2,166,300 |
| 1,000,000,000 | 6,000,000,001 | 279,545,369 | 17,244,408 |
| 10,000,000,000 | 60,000,000,001 | 2,524,038,155 | 140,494,396 |
| 100,000,000,000 | 600,000,000,001 | 23,007,501,786 | 1,166,916,932 |
| 1,000,000,000,000 | 6,000,000,000,001 | 211,381,427,039 | 9,846,842,483 |
| 10,000,000,000,000 | 60,000,000,000,001 | 1,955,010,428,258 | 84,209,699,420 |
| 100,000,000,000,000 | 600,000,000,000,001 | 18,184,255,291,570 | 728,412,916,122 |
| 1,000,000,000,000,000 | 6,000,000,000,000,001 | 169,969,662,554,552 | 6,363,082,106,476 |