A279161 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.

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, 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, 15, 45, 3, 46, 16, 22, 18, 34, 5, 51, 17, 29, 8, 54, 8, 56, 20, 23, 19

Offset: 3

Author

Vladimir Shevelev, Dec 07 2016

Keywords

• nonn

Comments

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.

References

• E. Landau, Handbuch der Lehre yon der Verteilung der Primzahlen, 2 vols., Leipzig, Teubner, 1909. Reprinted in 1953 by Chelsea Publishing Co., New York.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 3..5002
• J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp. 64-94.

Crossrefs

Cf. A000010, A001620.

Programs

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

Extensions

More terms from Peter J. C. Moses, Dec 07 2016