A343480 -id:A343480 - cp's OEIS frontend

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

Artur Jasinski, Feb 09 2025

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,....

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R. P. Brent, Tables of T[r,k] and A[r,k], (1970) [unpublished].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

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

(* 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]

A380948 Denominators of rational coefficients which are ratio of Brent's coefficients -A[n,2]/A343480.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 4, 8, 16, 2, 40, 32, 80, 20, 112, 1120, 320, 112, 112, 640, 32, 1120, 160, 5600, 280, 64, 1820, 116480, 2240, 14560, 232960, 3136, 364, 18200, 34, 116480, 618800, 76160, 10640, 1074944, 30464, 110656, 18811520, 13600, 2434432, 181060880, 15232, 3043040

Offset: 1

Artur Jasinski, Feb 09 2025

Brent's coefficients -A[n,2]/A343480 are rationals =  A380947(n)/ A380948(n).
Brent's coefficients are used in formulas of number of primes with particular distance to next prime =2*n.
Brent's coefficients A[n,1]/A114907 = B[n,1]/A114907 are equal to A380839(n)/A307410(n).

R. P. Brent, The distribution of small gaps between successive primes, Mathematics of Computation 28 (1974), 315-324.

Cf. A307410, A380839, A343480, A380947, A381083.

A343478 Prime numbers p == 2 (mod 3) such that p-1 has exactly one distinct odd prime divisor and p+1 has exactly one distinct prime divisor > 3.

Original entry on oeis.org

29, 41, 59, 83, 89, 101, 113, 137, 149, 167, 173, 179, 197, 227, 233, 251, 263, 269, 293, 317, 347, 353, 359, 401, 449, 467, 479, 503, 557, 563, 587, 593, 641, 653, 677, 719, 773, 809, 887, 977, 983, 1097, 1187, 1193, 1283, 1307, 1367, 1373, 1433, 1439, 1487, 1493

Offset: 1

Amiram Eldar, Apr 16 2021

nonn

Esparza and Gehring (2018) proved that assuming a generalized Hardy-Littlewood conjecture the number of terms not exceeding x is asymptotically (c/2) * x/log(x)^3, where c = A343480 = 5.716497...

			29 is a term since it is prime, 29 = 3*9 + 2, 29-1 = 28 = 2^2 * 7 has only one distinct odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one distinct prime divisor (5) larger than 3.
101 is a term since it is prime, 101 = 3*33 + 2, 101-1 = 100 = 2^2 * 5^2 has only one distinct odd prime divisor (5) and 101+1 = 102 = 2^2*3*17 has only one distinct prime divisor (17) larger than 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlos Esparza and Lukas Gehring, Estimating the density of a set of primes with applications to group theory, arXiv:1810.08679 [math.NT], 2018.

A343479 is a subsequence.

Cf. A003627, A215504, A336101, A343480.

q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeNu[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] == 1 && PrimeNu[(n - 1)/2^IntegerExponent[n - 1, 2]] == 1; Select[Range[1500], q]

A343479 Prime numbers p == 2 (mod 3) such that p-1 has exactly one odd prime divisor and p+1 has exactly one prime divisor > 3 (counting prime divisors with multiplicity in both).

Original entry on oeis.org

29, 41, 59, 83, 89, 113, 137, 167, 173, 179, 227, 233, 263, 269, 317, 347, 353, 359, 467, 479, 503, 557, 563, 593, 641, 653, 719, 773, 809, 887, 977, 983, 1097, 1187, 1193, 1283, 1307, 1367, 1433, 1439, 1487, 1493, 1523, 1619, 1697, 1823, 1907, 1997, 2063, 2153

Offset: 1

Amiram Eldar, Apr 16 2021

nonn

Esparza and Gehring (2018) proved that assuming a generalized Hardy-Littlewood conjecture the number of terms not exceeding x is asymptotically (c/2) * x/log(x)^3, where c = A343480 = 5.716497...

			29 is a term since it is prime, 29 = 3*9 + 2, 29-1 = 28 = 2^2 * 7 has only one odd prime divisor (7) and 29+1 = 30 = 2*3*5 has only one prime divisor (5) larger than 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carlos Esparza and Lukas Gehring, Estimating the density of a set of primes with applications to group theory, arXiv:1810.08679 [math.NT], 2018.

Subsequence of A343478.

Cf. A003627, A038550, A215504, A343480.

q[n_] := Mod[n, 3] == 2 && PrimeQ[n] && PrimeQ[(n + 1)/2^IntegerExponent[n + 1, 2]/3^IntegerExponent[n + 1, 3]] && PrimeQ[(n - 1)/2^IntegerExponent[n - 1, 2]]; Select[Range[2000], q]

cp's OEIS Frontend

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

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A380948 Denominators of rational coefficients which are ratio of Brent's coefficients -A[n,2]/A343480.

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A343478 Prime numbers p == 2 (mod 3) such that p-1 has exactly one distinct odd prime divisor and p+1 has exactly one distinct prime divisor > 3.

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Mathematica

A343479 Prime numbers p == 2 (mod 3) such that p-1 has exactly one odd prime divisor and p+1 has exactly one prime divisor > 3 (counting prime divisors with multiplicity in both).

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