A078559 - OEIS

%I #71 Dec 01 2024 11:34:14

%S 3,6,9,12,72,84,189,21,252,270,288,304,1596,152,3648,49248,295488,

%T 1526688,17302464,622888704,640191168,1707176448,10243058688,

%U 23046882048,23527025424,599939148312,47054050848,2540918745792

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

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

%H Robert Israel, <a href="/A078559/b078559.txt">Table of n, a(n) for n = 1..1701</a> (b-file corrected by _Georg Fischer_, Jan 16 2019)

%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.

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

%F 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

%p Q:= 1: p:= 1:

%p for n from 1 to 100 do

%p   p:= nextprime(p);

%p   Q:= Q * (p+1)/(p-1);

%p   A[n]:= numer(Q);

%p od:

%p seq(A[i],i=1..100); # _Robert Israel_, May 11 2018

%t Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* _Alonso del Arte_, Aug 23 2011 *)

%o (PARI) a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ _Michel Marcus_, May 11 2018

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

%Y Denominators are in A078560.

%K nonn,frac

%O 1,1

%A _Labos Elemer_, Dec 06 2002

%E Improved definition from _Franklin T. Adams-Watters_, Dec 02 2005