This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A048869 #13 Aug 10 2025 19:54:59
%S A048869 3,4,5,6,7,9,10,15,21,45,58,82,86,92,105,116,196,238,308,310,320,380,
%T A048869 972,978,996,1068,1368,5640,10890
%N A048869 Numbers for which reduced residue system contains as many primes as nonprimes.
%C A048869 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 A048869 There are no terms larger than 10890; it suffices to check to 52024. [_Charles R Greathouse IV_, Dec 19 2011]
%e A048869 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 A048869 Select[Range[700000], 2(PrimePi[ # ] - Length[FactorInteger[ # ]]) == EulerPhi[ # ]&]
%t A048869 For[i = 1, i < 20000, 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 A048869 (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 A048869 A000720(n)-A001221(n) = A000010(n) - [ A000720(n)-A001221(n) ].
%Y A048869 Cf. A048597-A002110.
%K A048869 nonn,fini,full
%O A048869 1,1
%A A048869 _Labos Elemer_
%E A048869 More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006