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
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!.
Crossrefs
Programs
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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
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Mathematica
Numerator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *) -
PARI
a(n) = numerator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018
Formula
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
Extensions
Improved definition from Franklin T. Adams-Watters, Dec 02 2005