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%I #17 Nov 17 2019 01:57:47
%S 1,1,3,1,4,2,4,2,7,1,9,3,4,4,12,2,13,3,7,5,17,2,14,6,12,5,21,1,23,9,
%T 12,8,16,4,27,9,15,7,31,2,32,10,14,12,35,5,31,9,20,12,40,6,28,11,23,
%U 15,45,3,46,16,22,18,34,5,51,17,29,8,54,8,56,20,23,19
%N Define P = e^gamma*loglog(n) and Q = 3/loglog(n), where gamma is Euler's constant A001620. Then a(n) = phi(n) - ceiling(n/(P + Q)), where phi(n) is Euler's function A000010.
%C The best known lower estimate for phi(n)is phi(n) > n/(P + Q), n >= 3 [Rosser and Schoenfeld] (and, for each eps > 0, there exist infinitely many n for which phi(n) < n/P', where in P' e^gamma is replaced by e^(gamma-eps) [Landau]). So a(n) >= 0.
%D E. Landau, Handbuch der Lehre yon der Verteilung der Primzahlen, 2 vols., Leipzig, Teubner, 1909. Reprinted in 1953 by Chelsea Publishing Co., New York.
%H Peter J. C. Moses, <a href="/A279161/b279161.txt">Table of n, a(n) for n = 3..5002</a>
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>. Illinois J. Math. 6 (1962), pp. 64-94.
%o (PARI) a(n)=my(LL=log(log(n)),P=LL*exp(Euler),Q=3/LL); eulerphi(n) - ceil(n/(P+Q)) \\ _Charles R Greathouse IV_, Dec 07 2016
%Y Cf. A000010, A001620.
%K nonn
%O 3,3
%A _Vladimir Shevelev_, Dec 07 2016
%E More terms from _Peter J. C. Moses_, Dec 07 2016
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