Enumeration of the prime quadruplets to 1e16 Thomas R. Nicely

http://www.trnicely.netCurrent e-mail address

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_4(x), the count of prime quadruplets (q, q+2, q+6, q+8) such that q <= x. The first three such quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), and (101, 103, 107, 109). Also provided are the values of the related functions delta_4(x), S_4(x), and F_4(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 quadruplets from 1e16 to 2e16.

Complete counts and reciprocal sums of the prime constellations from Nicely's computations (1993-2009), including the prime quadruplets, 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.

Symbols are defined as follows; for further details, see the paper Enumeration to $1.6 \times 10^{15}$ of the prime quadruplets.

• pi_4(x) = Number of prime quadruplets (q, q+2, q+6, q+8) such that the first element q <= x .
• delta_4(x) = Li_4(x) - pi_4(x) . A positive value indicates a deficit of quadruplets (compared to the theoretical estimate); a negative value indicates a surplus of quadruplets.
• Li_4(x) = integral(27/2*c_4/((ln(t))^4), t, 2, x) = Hardy-Littlewood integral approximation for pi_4(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_4*(x) = integral(27/2*c_4/((ln(t+6))^4), t, 2, x) .
• c_4 = Hardy-Littlewood quadruplets constant = 0.30749 48787 58327 09312 33544 86071 07685 3.... 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_4(x) = Sum of the reciprocals of all of the elements of the prime quadruplets (q, q+2, q+6, q+8) such that q <= x . Note that if x >= 11, the terms 1/11 and 1/13 are included twice.
• F_4(x) = First order extrapolation, from S_4(x), of the Brun's constant B_4 of the quadruplets (the limit of the sum of the reciprocals as x approaches +infinity): B_4 \approx F_4(x) = S_4(x) + 18c_4/(ln(x))^3 .
• The values given for pi_4(x) are believed to be exact. The values for delta_4(x), S_4(x), and F_4(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.