A380947 Numerators of rational coefficients which are ratio of Brent's coefficients -A[n,2]/A343480.

0, 0, 1, 1, 3, 7, 5, 5, 23, 39, 63, 17, 209, 185, 1207, 127, 765, 15543, 2499, 1139, 2257, 6327, 309, 21527, 2189, 64273, 6127, 883, 21681, 3835077, 30537, 188579, 7091843, 47895, 8447, 556651, 541, 1978953, 22046359, 1726463, 188751, 45916389, 575107, 2289527, 968180019, 283521, 50207679, 7450167293, 385389, 86547757

Offset: 1

Author

Artur Jasinski, Feb 09 2025

Keywords

• nonn
• frac

Comments

Brent's coefficients -A[n,2]/A343480 are rationals = A380947(n)/A380948(n).

Number of primes with distance to next prime = 2*n between two particular numbers j and k is ~ equal Integrate_{s,j,k} Sum_{m,1,m_max} A[n,m]/log(s)^(m+1).

Brent's coefficients A[n,1]/A114907 = B[n,1]/A114907 are equal to A380839(n)/A307410(n).

Real Brent's coefficients A[n,2] = -A343480*A380947(n)/A380948(n).

Integer Brent's coefficients T[n,2] = A381085(n).

Maximal values of the coefficients A380947(n)/A380948(n) occurs when n=105*k where k=1,2,3,4,....

Minimal values of the coefficients A380947(n)/A380948(n) occurs when n=2^k where k=0, 1,2,3,4,....

Links

• R. P. Brent, Empirical evidence for a proposed distribution of small prime gaps, Technical report NO. CS 123 (1969) Stanford University.
• R. P. Brent, Tables of T[r,k] and A[r,k], (1970) [unpublished].
• R. P. Brent, The distribution of small gaps between successive primes, Mathematics of Computation 28 (1974), 315-324, JSTOR
• Artur Jasinski, List of coefficients A380947(n)/A380948(n) for n <= 717

Crossrefs

Cf. A307410, A380839, A343480, A380948, A381085.

Programs

Mathematica

(* starting vector tr2 taken from A381085 *)
tr2 ={0, 0, 2, 4, 6, 56, 40, 40, 92, 624, 504, 10880, 6688, 7400, 19312};
ww = {}; long=15;Do[kk = PrimePi[n + 1]; prod = 1;
 Do[prod = prod (Prime[n] - 1), {n, 2, kk}];
 AppendTo[ww, prod], {n, 1, long}]; sr2 = {}; Do[
 AppendTo[sr2, tr2[[n]]/ww[[n]]], {n, 1, long}]; fr2 = {}; uu = {}; Do[
 pr1 = 1; kk = PrimePi[p + 1]; pr3 = 1;
 Do[pr2 = 1; jj = Min[2, Prime[n] - 2];
  Do[pr2 = pr2 (1 - m/((Prime[n] - 1) (Prime[n] - m))), {m, 1, jj}];
  pr1 = pr1 pr2; pr3 = pr3 Prime[n]/(Prime[n] - 1), {n, 2, kk}];
 pr3 = (-2 pr3)^2/pr1; AppendTo[fr2, pr3], {p, 1, long}]; ar2 = {}; Do[
 AppendTo[ar2, fr2[[n]] sr2[[n]]/12], {n, 1, long}]; Numerator[ar2]