cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A307410 Numerators of the product in the singular series.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 5, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 11, 9, 65, 15, 21, 5, 69, 1, 71, 35, 3, 17, 3, 11, 77, 3, 1, 39, 81, 5, 45
Offset: 1

Views

Author

Mats Granvik, Apr 07 2019

Keywords

Comments

Differs from A305444 at n = 35, 65, 70, ...

Links

Crossrefs

Cf. A005596, A005597, A305444, A380839 (denominators).

Programs

  • Maple
    f:= proc(n) numer(mul((p-2)/(p-1),p=select(type,numtheory:-factorset(n),odd))) end proc:
map(f, [$1..100]); # Robert Israel, Apr 07 2019
  • Mathematica
    Table[Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1],2]] - 2)/Times@@(DeleteDuplicates[DeleteCases[DeleteCases[Exp[MangoldtLambda[Divisors[h]]], 1], 2]] - 1), {h, 1, 85}]
Numerator[%]
f[p_, e_] := If[p == 2, 1, (p-2)/(p-1)]; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 03 2025 *)
  • PARI
    a(n) = my(f=factor(n)[,1]~); numerator(prod(k=1, #f, if (f[k]>2, (f[k]-2)/(f[k]-1), 1))); \\ Michel Marcus, Apr 07 2019

Formula

a(n) = numerator of Product_{p|n;p>2}(p-2)/(p-1) where p is a prime number.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A380839(k) = 2 * Product_{p prime} (1-1/(p^2-p)) = 2 * A005596 = 0.7479116272384045761094... . - 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

Views

Author

Artur Jasinski, Feb 09 2025

Keywords

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

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]

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

Views

Author

Artur Jasinski, Feb 09 2025

Keywords

Comments

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).

Links

Crossrefs

Cf. A307410, A380839, A343480, A380947, A381083.
Showing 1-3 of 3 results.

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