A209079 -id:A209079 - cp's OEIS frontend

3, 4, 13, 67, 560, 6095, 87693, 1491707, 30942952, 795721368, 22614834943, 759296069174, 28510284114397, 1148788714239052, 50932190960133487, 2532582753383324327, 139681393339880282191, 8089483267352888074399, 512986500081861276401709, 34658318003703434434962860

Offset: 1

Jonathan Sondow, Sep 29 2012

nonn

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.
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.
See A185339 for additional links, references, and formulas.
Named after the French mathematician Jean-Louis Nicolas. - Amiram Eldar, Jun 23 2021

			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.

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.

Amiram Eldar, Table of n, a(n) for n = 1..350
J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory, Vol. 17, No.3 (1983), pp. 375-388.
J.-L. Nicolas, Small values of the Euler function and the Riemann hypothesis, arXiv:1202.0729 [math.NT], 2012; Acta Arith., Vol. 155 (2012), pp. 311-321.

Cf. A000010, A001620, A002110, A005867, A185339, A209079, A218245, A233825.

primorial[n_] := Product[Prime[k], {k, n}]; Table[With[{p = primorial[n]}, p - Floor[EulerPhi[p]*Log[Log[p]]*Exp[EulerGamma]]], {n, 1, 20}]

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

A216868(n)={(n=prod(i=1,n,prime(i)))-floor(eulerphi(n)*log(log(n))*exp(Euler))}  \\ M. F. Hasler, Oct 06 2012

a(n) = prime(n)# - floor(phi(prime(n)#)*log(log(prime(n)#))*e^gamma).
a(n) = A002110(n) - floor(A005867(n)*log(log(A002110(n)))*e^gamma).
Limit_{n->oo} a(n)/p(n)# = 0.

cp's OEIS Frontend

A216868 Nicolas's sequence whose positivity is equivalent to the Riemann hypothesis.

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