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%I #26 Oct 21 2016 13:46:45
%S 138594,249474,277194,471234,554394,665274,900870,1015554,1081074,
%T 1191954,1244874,1358274,1385994,1607754,1801794,1857234,2189874,
%U 2356170,2356194,2411634,2439354,2489754,2522514,2550234,2633394,2688834,2702670,2716554
%N Numbers n such that phi(sigma(n)) + sigma(phi(n)) < n.
%C Let ps(n) be number of terms of the sequence up to n, it seems that ps(n) ~ n/100000. Is it true that 6 divides each term of the sequence?
%C I guess that there is no number n such that phi(sigma(n)) + sigma(phi(n)) = n.
%C From _M. F. Hasler_, Oct 31 2013: (Start)
%C Most terms of the sequence are of the form given in the following
%C Theorem: If p is a safe prime (A005385), then n = 6p is a term of this sequence if and only if (1-1/q1)*...*(1-1/qr) + 7/12 < p/(p+1), where q1,...,qr are the distinct odd prime factors of p+1.
%C Proof: Write p+1 = 2^a 3^b Q with gcd(Q,6)=1 and assume (p-1)/2 is prime. For n = 6p, an easy calculation yields phi(sigma(n)) + sigma(phi(n)) = n*(1+1/p)*(2/3*(1-1/q2)*...*(1-1/qr)+7/12), where q2,...,qr are the prime factors of Q. #
%C Corollary: n=6p is in the sequence when p is a safe prime and p+1 is a multiple of 2*3*5*7*11 or of 2*3*5*7*13*q with some prime q>13, q<80. (End)
%H Donovan Johnson, <a href="/A230032/b230032.txt">Table of n, a(n) for n = 1..1000</a>
%t Do[If[EulerPhi[DivisorSigma[1,n]] + DivisorSigma[1,EulerPhi[n]] < n, Print[n]], {n,3300000}]
%o (PARI) is_A230032(n)={eulerphi(sigma(n))+sigma(eulerphi(n))<n} \\ _M. F. Hasler_, Oct 31 2013
%Y Cf. A000010, A000203, A062401, A062402, A066950.
%K nonn
%O 1,1
%A _Farideh Firoozbakht_, Oct 28 2013
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