cp's OEIS Frontend

From Vladimir Shevelev, Dec 08 2016: (Start)

Define P = exp(gamma)*log(log(k)), where gamma is Euler's constant A001620. The sequence lists numbers k for which phi(k) < k/P, where phi(k) is Euler's function A000010.

In 1909, Landau proved that for each eps>0, there exist infinitely many k for which phi(k) < k/P', where P' = exp(gamma-eps)*log(log(k)). In 1983 Nicolas strengthened Landau's result showing that there exist infinitely many k for which phi(k) < k/P. So this sequence is infinite.

All terms are even, except for 3,5,9 and 15. See proof in [Choie et al., Theorem 2.1]. (End)

Assuming the Riemann hypothesis, no term exceeds 4. Indeed, let c(n) = (n/phi(n) - (e^gamma)*loglog(n))*sqrt(log(n)). Then, by [Nicolas], the Riemann hypothesis is equivalent to the inequality: for n>=2, c(n)<=c(N), where N is the product of the first 66 primes such that c(N)=4.0628356921... . Since for n in [or "not in", the grammar of the original was ambiguous here - N. J. A. Sloane, Jan 04 2017] A100966, we have c(n)<=0, for those n c(n)<=c(N). Thus assuming the R. H. we see that a(n)<=4.

On the other hand, we conjecture that a(n)<=4 should be true independent of the R. H. If so, then the statement that the R. H. is false would be equivalent to the existence of n for which c(n) is in interval (c(N),5).