## Enumeration of the prime triplets (q, q+2, q+6) to 1e16 Thomas R. Nicely

Freeware copyright (c) 2010 Thomas R. Nicely. Released into the public domain by the author, who disclaims any legal liability arising from its use.

Last updated 1000 GMT 18 January 2010.

This is a table of values of pi_3a(x), the count of prime triplets (q, q+2, q+6) such that q <= x. The first three such triplets are (5, 7, 11), (11, 13, 17), and (17, 19, 23). Also provided are the values of the related functions delta_3a(x), S_3a(x), and F_3a(x). The counts in this table were obtained by a direct and explicit generation and enumeration of the primes.

The domain of this table consists of each decade from 10 through 1e12, then each 1e12 to 1e16. See also Enumeration of the prime triplets (q, q+2, q+6) from 1e16 to 2e16.

Complete counts and reciprocal sums of the prime constellations from Nicely's computations (1993-2009), including the prime triplets, are also available. These data files are very large (over 60MB each, even for the zipped versions), including more than two million data points from 0 to 2e16 at intervals of 1e10 or better.

The symbols are defined as follows.

• pi_3a(x) = Number of prime triplets (q, q+2, q+6) such that the first element q <= x .
• delta_3a(x) = Li_3a(x) - pi_3a(x) . A positive value indicates a deficit of such triplets (compared to the theoretical estimate); a negative value indicates a surplus of these triplets.
• Li_3a(x) = integral(9/2*c_3/((ln(t))^3), t, 2, x) = Hardy-Littlewood integral approximation for pi_3a(x) . Although this is the traditional formula, note that a slightly more accurate (for small x) approximation is produced by the asymptotically equivalent formula Li_3a*(x) = integral(9/2*c_3/((ln(t+10))^3), t, 2, x) .
• c_3 = Hardy-Littlewood triplets constant = 0.63516 63546 04271 20720 66965 91272 52241 7342.... The kth Hardy-Littlewood constant (k > 1) is defined as c_k = prod((p^(k-1))*(p-k)/(p-1)^k, p; p prime, p > k) .
• S_3a(x) = Sum of the reciprocals of all of the elements of the prime triplets (q, q+2, q+6) such that q <= x . Note that if x >= 11, the term 1/11 is included twice; if x >= 17, the term 1/17 is included twice; if x >= 107, the term 1/107 is included twice; if x > = 1487, the term 1/1487 is included twice; and so on.
• F_3a(x) = First order extrapolation, from S_3a(x), of the Brun's constant B_3a of the (q, q+2, q+6) triplets (the limit of the sum of the reciprocals as x approaches +infinity): B_3a \approx F_3a(x) = S_3a(x) + 27/4*c_3/(ln(x))^2 .
• The values given for pi_3a(x) are believed to be exact. The values for delta_3a(x), S_3a(x), and F_3a(x) are believed to be correct to all digits shown, except for a possible (rounding) error of one ulp (one unit in the last decimal place or least significant digit).
• Please inform me of any errors you find in these values.