

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
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OFFSET

1,1


COMMENTS

a(n) = floor(p(n)#/phi(p(n)#)  log(log(p(n)#))*exp(gamma)), where p(n)# is the nth 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 crossrefs.


LINKS

Table of n, a(n) for n=1..97.


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] = [30.58319...*1.78107...] = [31.038...] = 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



