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

%I A078559 #74 Apr 16 2025 05:27:45
%S A078559 3,6,9,12,72,84,189,21,252,270,288,304,1596,152,3648,49248,295488,
%T A078559 1526688,17302464,622888704,640191168,1707176448,10243058688,
%U A078559 23046882048,23527025424,599939148312,47054050848,2540918745792
%N A078559 Numerator of Product_{i=1..n} (p_i + 1)/(p_i - 1) where p_i is the i-th prime.
%D A078559 R. K. Guy, Unsolved Problems in Number Theory, B48.
%H A078559 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 A078559 F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.
%F A078559 a(n) = A054640(n)/A078558(n).
%F A078559 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} arctanh(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 A078559 Q:= 1: p:= 1:
%p A078559 for n from 1 to 100 do
%p A078559   p:= nextprime(p);
%p A078559   Q:= Q * (p+1)/(p-1);
%p A078559   A[n]:= numer(Q);
%p A078559 od:
%p A078559 seq(A[i],i=1..100); # _Robert Israel_, May 11 2018
%t A078559 Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* _Alonso del Arte_, Aug 23 2011 *)
%o A078559 (PARI) a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ _Michel Marcus_, May 11 2018
%Y A078559 Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A091724, A013661.
%Y A078559 Denominators are in A078560.
%K A078559 nonn,frac
%O A078559 1,1
%A A078559 _Labos Elemer_, Dec 06 2002
%E A078559 Improved definition from _Franklin T. Adams-Watters_, Dec 02 2005