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%I #20 Dec 07 2015 22:40:32
%S 2,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N Nicolas's sequence, whose nonnegativity is equivalent to the Riemann hypothesis.
%C a(n) = floor(p(n)#/phi(p(n)#) - log(log(p(n)#))*exp(gamma)), where p(n)# is the n-th primorial, phi is Euler's totient function, and gamma is Euler's constant.
%C J.-L. Nicolas proved that all terms are >= 0 if and only if the Riemann hypothesis (RH) is true. In fact, results in his 2012 paper imply that RH is equivalent to a(n) = 0 for n > 6. Nicolas's refinement of this result is in A233825.
%C He also proved that if RH is false, then infinitely many terms are >= 0 and infinitely many terms are < 0.
%C See Nicolas's sequence A216868 for references, links, and additional cross-refs.
%F a(n) = [p(n)#/phi(p(n)#) - log(log(p(n)#))*exp(gamma)].
%F a(n) = [A002110(n)/A005867(n) - log(log(A002110(n)))*e^gamma].
%e p(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = [6/2 - log(log(6))*e^gamma] = [3-0.58319...*1.78107...] = [3-1.038...] = 1.
%t primorial[n_] := Product[Prime[k], {k, n}]; Table[ With[{p = primorial[n]}, Floor[N[p/EulerPhi[p] - Log[Log[p]]*Exp[EulerGamma]]]], {n, 1, 100}]
%Y Cf. A216868, A233825.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Oct 24 2012
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