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%I #10 Oct 30 2022 18:19:59
%S 3,4,5,6,7,9,10,15,21,45,58,82,86,92,105,116,196,238,308,310,320,380,
%T 972,978,996,1068,1368,5640,10890
%N Numbers for which reduced residue system contains as many primes as nonprimes.
%C This sequence is finite, since the number of primes < n is ~ n/log(n), but liminf phi(n) / ( n*log(log(n)) ) = exp(-gamma), a consequence of Mertens's theorem (see Hardy and Wright's Theory of Numbers). Also, if there exists a further element, it is >700000 (as verified with the enclosed Mathematica code). (Question: is it possible to show that there are no further such elements by using explicit bounds in the Prime Number Theorem and in Mertens's theorem?) - _Reiner Martin_, Jan 16 2002
%C There are no terms larger than 10890; it suffices to check to 52024. [_Charles R Greathouse IV_, Dec 19 2011]
%e n=45, phi(45)=24 and the reduced residue system mod 45 contains 12 primes {2,7,11,13,17,19,23,29,31,37,41,43} and 12 nonprimes {1,4,8,14,16,22,26,28,32,34,38,44}.
%t Select[Range[700000], 2(PrimePi[ # ] - Length[FactorInteger[ # ]]) == EulerPhi[ # ]&]
%t For[i = 1, i < 100000000000, i++, If[2(PrimePi[i] - Length[FactorInteger[i]]) == EulerPhi[i], Print[i]]]; - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006
%o (PARI) p=0;for(n=1,6e4,if(isprime(n),p++);if(p==eulerphi(n)/2+omega(n),print1(n", "))) \\ _Charles R Greathouse IV_, Dec 19 2011
%Y A000720(n)-A001221(n) = A000010(n) - [ A000720(n)-A001221(n) ].
%Y Cf. A048597-A002110.
%K nonn,fini,full
%O 1,1
%A _Labos Elemer_
%E More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006
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