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%I #52 Nov 07 2023 03:15:57
%S 3,4,13,67,560,6095,87693,1491707,30942952,795721368,22614834943,
%T 759296069174,28510284114397,1148788714239052,50932190960133487,
%U 2532582753383324327,139681393339880282191,8089483267352888074399,512986500081861276401709,34658318003703434434962860
%N Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis.
%C a(n) = p(n)# - floor(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 All a(n) are > 0 if and only if the Riemann hypothesis is true. If the Riemann hypothesis is false, then infinitely many a(n) are > 0 and infinitely many a(n) are <= 0. Nicolas (1983) proved this with a(n) replaced by p(n)#/phi(p(n)#)-log(log(p(n)#))*exp(gamma). Nicolas's refinement of this result is in A233825.
%C See A185339 for additional links, references, and formulas.
%C Named after the French mathematician Jean-Louis Nicolas. - _Amiram Eldar_, Jun 23 2021
%D J.-L. Nicolas, Petites valeurs de la fonction d'Euler et hypothèse de Riemann, in Seminar on Number Theory, Paris 1981-82 (Paris 1981/1982), Birkhäuser, Boston, 1983, pp. 207-218.
%H Amiram Eldar, <a href="/A216868/b216868.txt">Table of n, a(n) for n = 1..350</a>
%H J.-L. Nicolas, <a href="http://math.univ-lyon1.fr/~nicolas/petitsphi83.pdf">Petites valeurs de la fonction d'Euler</a>, J. Number Theory, Vol. 17, No.3 (1983), pp. 375-388.
%H J.-L. Nicolas, <a href="http://arxiv.org/abs/1202.0729">Small values of the Euler function and the Riemann hypothesis</a>, arXiv:1202.0729 [math.NT], 2012; Acta Arith., Vol. 155 (2012), pp. 311-321.
%F a(n) = prime(n)# - floor(phi(prime(n)#)*log(log(prime(n)#))*e^gamma).
%F a(n) = A002110(n) - floor(A005867(n)*log(log(A002110(n)))*e^gamma).
%F Limit_{n->oo} a(n)/p(n)# = 0.
%e prime(2)# = 2*3 = 6 and phi(6) = 2, so a(2) = 6 - floor(2*log(log(6))*e^gamma) = 6 - floor(2*0.58319...*1.78107...) = 6 - floor(2.07...) = 6 - 2 = 4.
%t primorial[n_] := Product[Prime[k], {k, n}]; Table[With[{p = primorial[n]}, p - Floor[EulerPhi[p]*Log[Log[p]]*Exp[EulerGamma]]], {n, 1, 20}]
%o (PARI) nicolas(n) = {p = 2; pri = 2;for (i=1, n, print1(pri - floor(eulerphi(pri)*log(log(pri))*exp(Euler)), ", ");p = nextprime(p+1);pri *= p;);} \\ _Michel Marcus_, Oct 06 2012
%o (PARI) A216868(n)={(n=prod(i=1,n,prime(i)))-floor(eulerphi(n)*log(log(n))*exp(Euler))} \\ _M. F. Hasler_, Oct 06 2012
%Y Cf. A000010, A001620, A002110, A005867, A185339, A209079, A218245, A233825.
%K nonn
%O 1,1
%A _Jonathan Sondow_, Sep 29 2012
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