A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.

3, 6, 9, 12, 72, 84, 189, 21, 252, 270, 288, 304, 1596, 152, 3648, 49248, 295488, 1526688, 17302464, 622888704, 640191168, 1707176448, 10243058688, 23046882048, 23527025424, 599939148312, 47054050848, 2540918745792

Offset

1,1

References

R. K. Guy, Unsolved Problems in Number Theory, B48.

Links

Robert Israel, Table of n, a(n) for n = 1..1701 (b-file corrected by Georg Fischer, Jan 16 2019)

F. Smarandache, Only Problems, Not Solutions!.

Formula

a(n) = A054640(n)/A078558(n).

a(n)/A078560(n) ~ C*log^2(prime(n)), where C = exp(2*gamma)/zeta(2) = 6(e^gamma/pi)^2 = A091724 / A013661. Physics note: (a(n)/A078560(n) - 1)/(a(n)/A078560(n) + 1) = tanh(Sum_{k=1..n} artanh(1/prime(k))) is the relativistic sum of n velocities c/2, c/3, ..., c/prime(n), in units where the speed of light c = 1. - Thomas Ordowski, Nov 06 2024

Maple

Q:= 1: p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  Q:= Q * (p+1)/(p-1);
  A[n]:= numer(Q);
od:
seq(A[i], i=1..100); # Robert Israel, May 11 2018

Mathematica

Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *)

Prog

(PARI) a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018

Crossrefs

Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A091724, A013661.

Denominators are in A078560.

Sequence in context: A072179 A065811 A061514 * A317557 A338763 A159908

Adjacent sequences: A078556 A078557 A078558 * A078560 A078561 A078562

Keyword

nonn,frac

Author

Labos Elemer, Dec 06 2002

Extensions

Improved definition from Franklin T. Adams-Watters, Dec 02 2005

Status

approved