A307410 -id:A307410 - cp's OEIS frontend

A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 8, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 16, 20, 66, 16, 44, 8, 70, 2, 72, 36

Offset: 1

Artur Jasinski, Feb 05 2025

This sequence is similar to A173557 but differences occurs for indices n=35,65,70,...
Coefficients J(n)=a(n)/A307410(n) occurs in many formulas on density of primes with gap 2*n.
Sylvester was the first who uses these coefficients at 1871.

			1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, ...
a(35) = 8 because 35 = 5 * 7 and then product is ((5-1)/(5-2))*((7-1)/(7-2)) = 8/5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70 (e.g. formula 4.11 p.32).
J. J. Sylvester, On the partition of an even number into two primes, Proc. London Math. Soc., ser. I, vol.4 (1871), pp. 4-6 (Math. Papers, vol. 2, pp. 709-711).
Marek Wolf, Applications of statistical mechanics in number theory, Physica A 274 (1999) 149-157 (formula (9) p. 154).

Cf. A167864, A173557, A305444, A307410 (denominators).

j = {}; Do[prod = 1; Do[If[PrimeQ[n] && IntegerQ[d/n], prod = prod (n - 1)/(n - 2)], {n, 3, d}]; AppendTo[j, prod], {d, 1, 74}]; Numerator[j]
f[p_, e_] := If[p == 2, 1, (p-1)/(p-2)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)

a(n) = my(f=factor(n)[,1]); numerator(prod(k=1, #f, if ((p=f[k])>2, (p-1)/(p-2), 1))); \\ Michel Marcus, Feb 05 2025

a(n) = numerator(A173557(n)/A305444(n)).
a(p^n) = p - 1 for prime p.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A307410(k) = Product_{prime p > 2} (1 + 1/(p*(p-2))) = 1.51478012... (A167864). - Amiram Eldar, Mar 03 2025

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

Original entry on oeis.org

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.

cp's OEIS Frontend

A380839 Numerators of J(n) = Product_{p|n, p odd prime} (p - 1)/(p - 2).

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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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