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

%I A380947 #53 Mar 02 2025 23:38:13
%S A380947 0,0,1,1,3,7,5,5,23,39,63,17,209,185,1207,127,765,15543,2499,1139,
%T A380947 2257,6327,309,21527,2189,64273,6127,883,21681,3835077,30537,188579,
%U A380947 7091843,47895,8447,556651,541,1978953,22046359,1726463,188751,45916389,575107,2289527,968180019,283521,50207679,7450167293,385389,86547757
%N A380947 Numerators of rational coefficients which are ratio of Brent's coefficients -A[n,2]/A343480.
%C A380947 Brent's coefficients -A[n,2]/A343480 are rationals = A380947(n)/A380948(n).
%C A380947 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).
%C A380947 Brent's coefficients A[n,1]/A114907 = B[n,1]/A114907 are equal to A380839(n)/A307410(n).
%C A380947 Real Brent's coefficients A[n,2] = -A343480*A380947(n)/A380948(n).
%C A380947 Integer Brent's coefficients T[n,2] = A381085(n).
%C A380947 Maximal values of the coefficients A380947(n)/A380948(n) occurs when n=105*k where k=1,2,3,4,....
%C A380947 Minimal values of the coefficients A380947(n)/A380948(n) occurs when n=2^k where k=0, 1,2,3,4,....
%H A380947 R. P. Brent, <a href="https://apps.dtic.mil/sti/pdfs/AD0696982.pdf">Empirical evidence for a proposed distribution of small prime gaps</a>, Technical report NO. CS 123 (1969) Stanford University.
%H A380947 R. P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pd/rpb021t.pdf">Tables of T[r,k] and A[r,k]</a>, (1970) [unpublished].
%H A380947 R. P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pub/pub021.html">The distribution of small gaps between successive primes</a>, Mathematics of Computation 28 (1974), 315-324, <a href="https://doi.org/10.2307/2005842">JSTOR</a>
%H A380947 Artur Jasinski, <a href="/A380947/a380947.dat.txt">List of coefficients A380947(n)/A380948(n) for n <= 717</a>
%t A380947 (* starting vector tr2 taken from A381085 *)
%t A380947 tr2 ={0, 0, 2, 4, 6, 56, 40, 40, 92, 624, 504, 10880, 6688, 7400, 19312};
%t A380947 ww = {}; long=15;Do[kk = PrimePi[n + 1]; prod = 1;
%t A380947  Do[prod = prod (Prime[n] - 1), {n, 2, kk}];
%t A380947  AppendTo[ww, prod], {n, 1, long}]; sr2 = {}; Do[
%t A380947  AppendTo[sr2, tr2[[n]]/ww[[n]]], {n, 1, long}]; fr2 = {}; uu = {}; Do[
%t A380947  pr1 = 1; kk = PrimePi[p + 1]; pr3 = 1;
%t A380947  Do[pr2 = 1; jj = Min[2, Prime[n] - 2];
%t A380947   Do[pr2 = pr2 (1 - m/((Prime[n] - 1) (Prime[n] - m))), {m, 1, jj}];
%t A380947   pr1 = pr1 pr2; pr3 = pr3 Prime[n]/(Prime[n] - 1), {n, 2, kk}];
%t A380947  pr3 = (-2 pr3)^2/pr1; AppendTo[fr2, pr3], {p, 1, long}]; ar2 = {}; Do[
%t A380947  AppendTo[ar2, fr2[[n]] sr2[[n]]/12], {n, 1, long}]; Numerator[ar2]
%Y A380947 Cf. A307410, A380839, A343480, A380948, A381085.
%K A380947 nonn,frac
%O A380947 1,5
%A A380947 _Artur Jasinski_, Feb 09 2025