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 A218245 Nicolas's sequence, whose nonnegativity is equivalent to the Riemann hypothesis. 2
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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. He also proved that if RH is false, then infinitely many terms are >= 0 and infinitely many terms are < 0. See Nicolas's sequence A216868 for references, links, and additional cross-refs. LINKS FORMULA a(n) = [p(n)#/phi(p(n)#) - log(log(p(n)#))*exp(gamma)]. a(n) = [A002110(n)/A005867(n) - log(log(A002110(n)))*e^gamma]. EXAMPLE 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. MATHEMATICA 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}] CROSSREFS Cf. A216868, A233825. Sequence in context: A165105 A325674 A055641 * A086075 A316865 A325616 Adjacent sequences:  A218242 A218243 A218244 * A218246 A218247 A218248 KEYWORD nonn AUTHOR Jonathan Sondow, Oct 24 2012 STATUS approved

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Last modified June 17 22:14 EDT 2021. Contains 345086 sequences. (Running on oeis4.)