

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.


0



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
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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



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, up1], T[[#]] < T[[# + 1]] &] (* Giovanni Resta, Mar 29 2017 *)


CROSSREFS



KEYWORD

nonn


AUTHOR

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


EXTENSIONS



STATUS

approved



