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%I #11 Mar 29 2017 11:30:19
%S 6,30,42,60,90,210,420,630,840,1050,2310,2730,3570,4620,5460,6930,
%T 9240,11550,13860,30030,39270,43890,60060,90090,120120,150150,180180,
%U 210210,240240,510510,570570,690690,1021020,1141140,1531530,1711710,2042040,2282280
%N 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.
%C 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.
%C 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
%H C. Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_033.htm">Perry's conjecture</a>
%e For all n > 90090, phi(n)/pi(n) > 2 > 17280/8726 = 1.9803.
%t 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 *)
%K nonn
%O 1,1
%A Luke Pebody (ltp1000(AT)hermes.cam.ac.uk), Feb 13 2003
%E Data corrected and extended by _Giovanni Resta_, Mar 29 2017
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