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A243996 Numbers n such that phi(sigma*(n)) = sigma*(phi(n)), where sigma*(n) is the sum of anti-divisors of n and phi(n) is the Euler totient function.

%I A243996 #22 Sep 28 2015 03:31:24
%S A243996 7,9,20,25,80,143,825,3117,3216,22774,52026,55804,138276,187733,
%T A243996 228384,265545,320766,549540,830814,839784,901376,1293552,1315776,
%U A243996 2635866,6771114,11126800,12087848,24351460,49382242,52344292,60063744,65980038,78279016,97638080
%N A243996 Numbers n such that phi(sigma*(n)) = sigma*(phi(n)), where sigma*(n) is the sum of anti-divisors of n and phi(n) is the Euler totient function.
%C A243996 a(70) > 10^10. - _Hiroaki Yamanouchi_, Sep 28 2015
%H A243996 Hiroaki Yamanouchi, <a href="/A243996/b243996.txt">Table of n, a(n) for n = 1..69</a>
%e A243996 sigma*(phi(25)) = sigma*(20) = 24, phi(sigma*(25)) = phi(39) = 24.
%p A243996 with(numtheory): P:=proc(q) local a,b,c,d,j,k,n;
%p A243996 for n from 1 to q do
%p A243996 k:=0; j:=n; while j mod 2<>1 do k:=k+1; j:=j/2; od;
%p A243996 a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;
%p A243996 k:=0; c:=phi(n); j:=phi(n); while j mod 2<>1 do k:=k+1; j:=j/2; od;
%p A243996 b:=sigma(2*c+1)+sigma(2*c-1)+sigma(c/2^k)*2^(k+1)-6*c-2;
%p A243996 if b=phi(a) then print(n); fi; od; end: P(10^10);
%t A243996 antiDivisors[n_] := Select[ Union[ Join[ Select[ Divisors[2 n - 1], OddQ[#] && # != 1 &], Select[ Divisors[ 2n + 1], OddQ[#] && # != 1 &], 2n/Select[ Divisors[ 2n], OddQ[#] && # != 1 &]]], # < n &]; fQ[n_] := EulerPhi@ Total@ antiDivisors@ n == Total@ antiDivisors@ EulerPhi@ n; k = 3; lst = {}; While[k < 10000001, If[ fQ@ k, AppendTo[lst, k]]; k++]; lst (* _Robert G. Wilson v_, Jun 21 2014 *)
%Y A243996 Cf. A000203, A066417, A230373, A033632.
%K A243996 nonn
%O A243996 1,1
%A A243996 _Paolo P. Lava_, Jun 18 2014
%E A243996 a(22)-a(25) from _Robert G. Wilson v_, Jun 21 2014
%E A243996 a(26)-a(34) from _Hiroaki Yamanouchi_, Sep 28 2015