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A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

%I A227011 #34 Jul 30 2025 19:28:06
%S A227011 1,3,5,11,13,17,29,41,181,209,377,779,3239,4469,5249,15539,43259,
%T A227011 58589,119279,169679,174719,461369,692687,955499,1258949,1859129,
%U A227011 1917299,3925463,7991693,8986469,13244069,16732169,30629363,44137523,48466987,64018433,68787773
%N A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) >  phi(sigma(m))/sigma(phi(m)) for all k<m.
%C A227011 These are the indices where the rational function A062401(n)/A062402(n) drops below the minimum set by all earlier ratios.
%C A227011 a(2) to a(9) are primes. However all known terms beginning from a(10) are composite.
%H A227011 Donovan Johnson, <a href="/A227011/b227011.txt">Table of n, a(n) for n = 1..48</a> (terms < 10^10)
%e A227011 5 is in the sequence because phi(sigma(5))/sigma(phi(5)) = 2/7 and for all k < 5, phi(sigma(k))/sigma(phi(k)) > 2/7.
%p A227011 A062401 := proc(n)
%p A227011     numtheory[phi](numtheory[sigma](n))
%p A227011 end proc:
%p A227011 A062402 := proc(n)
%p A227011     numtheory[sigma](numtheory[phi](n))
%p A227011 end proc:
%p A227011 s := proc(n)
%p A227011     A062401(n)/A062402(n) ;
%p A227011 end proc:
%p A227011 r := 100000000000000000000000000000 ;
%p A227011 for n from 1 do
%p A227011     if s(n) < r then
%p A227011         printf("%d,\n",n) ;
%p A227011         r := s(n) ;
%p A227011     end if;
%p A227011 end do:
%o A227011 (PARI) f(n)=eulerphi(sigma(n=factor(n)))/sigma(eulerphi(n))
%o A227011 is(n)=my(t=f(n)); for(k=1,n-1,if(f(k)<=t, return(0))); 1 \\ _Charles R Greathouse IV_, Nov 27 2013
%Y A227011 Cf. A227927, A062401, A062402, A033632, A229238.
%K A227011 nonn
%O A227011 1,2
%A A227011 _Vladimir Letsko_, Oct 09 2013
%E A227011 a(33)-a(37) from _Donovan Johnson_, Oct 11 2013