A080289 Integers n for which the ratio phi(n)/pi(n) is smaller than for any subsequent n. Here phi(n) is Euler's totient function and pi(n) is the number of primes that are at most n.

6, 30, 42, 60, 90, 210, 420, 630, 840, 1050, 2310, 2730, 3570, 4620, 5460, 6930, 9240, 11550, 13860, 30030, 39270, 43890, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 510510, 570570, 690690, 1021020, 1141140, 1531530, 1711710, 2042040, 2282280

Offset

1,1

Comments

Discovered while proving that phi(n) > 2pi(n) for all n, to prove a conjecture on www.primepuzzles.net. The conjecture stated that for all sufficiently large even n, n is the sum of two coprime composite numbers. This is in fact true for all even n > 210. For more details email the author.

Terms from 120120 to 2282280 have been determined empirically, by examining the ratio phi(k)/pi(k) for k up to 10^9. - Giovanni Resta, Mar 29 2017

Links

Table of n, a(n) for n=1..38.

C. Rivera, Perry's conjecture

Example

For all n > 90090, phi(n)/pi(n) > 2 > 17280/8726 = 1.9803.

Mathematica

up = 210211; f[n_] := If[n==1, 0, EulerPhi[n]/PrimePi[n]]; T = Array[f, up]; Do[T[[k]] = Min[T[[k]], T[[k + 1]]], {k, up - 1, 2, -1}]; Select[ Range[2, up-1], T[[#]] < T[[# + 1]] &] (* Giovanni Resta, Mar 29 2017 *)

Crossrefs

Sequence in context: A101939 A290466 A293188 * A175907 A114649 A090126

Adjacent sequences: A080286 A080287 A080288 * A080290 A080291 A080292

Keyword

nonn

Author

Luke Pebody (ltp1000(AT)hermes.cam.ac.uk), Feb 13 2003

Extensions

Data corrected and extended by Giovanni Resta, Mar 29 2017

Status

approved