Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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