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%I #36 Sep 08 2022 08:45:31
%S 8,16,64,256,16384,262144,1048576,4294967296
%N Numbers n such that sigma(phi(n)) -phi(n) -1 = phi(sigma(n) -n -1).
%C Used sigma(n)-n-1, namely the sum of proper divisors minus 1.
%C a(8) > 10^8. - _Michel Marcus_, Nov 01 2014
%C Every 2^(A000043+1) is a term. Proof sketch: Let ch=A048050 and n=2^k, then ch(phi(2^k))=phi(ch(2^k)), ch(2^(k-1))=phi(2^k-2), 2^(k-1)-2=phi(2^(k-1)-1), since phi(prime)=prime-1 the condition is satisfied by every k=A000043+1 or n=2^(A000043+1). See link. - _Jon Maiga_, Dec 14 2018
%C Conjecture: a(n)=2^(A000043(n)+1), if true the next terms are: 4294967296, 4611686018427387904, 1237940039285380274899124224... - _Jon Maiga_, Dec 14 2018
%C a(9) > 6.5*10^11. - _Giovanni Resta_, Dec 01 2019
%H Jon Maiga, <a href="http://jonkagstrom.com/eulerphi-chowla-mersenne/index.html">Euler Phi, Chowla and Mersenne primes</a>, 2018.
%p with(numtheory); P:=proc(n) local a,i; for i from 1 to n do
%p a:=phi(sigma(i)-i-1); if a>0 then
%p if sigma(phi(i))-phi(i)-1=a then print(i);
%p fi; fi; od; end: P(10^7);
%t ch[n_]:=DivisorSigma[1,n]-n-1
%t test[n_]:=ch[n]!=0 && ch[EulerPhi[n]] == EulerPhi[ch[n]]
%t Flatten[Position[Range[300000], Integer_ ? test]] (* _Jon Maiga_, Dec 14 2018 *)
%o (PARI) isok(n) = ((s=(sigma(n)-n-1)) != 0) && (sigma(eulerphi(n))-eulerphi(n)-1 == eulerphi(s)); \\ _Michel Marcus_, Nov 01 2014
%o (Magma) [n: n in [2..30000] | DivisorSigma(1,n) ne n+1 and DivisorSigma(1, EulerPhi(n)) - EulerPhi(n) - 1 eq EulerPhi(DivisorSigma(1,n) - n -1) ]; // _G. C. Greubel_, Dec 13 2018
%o (GAP) Filtered([4..1000000],n->Sigma(Phi(n))-Phi(n)-1=Phi(Sigma(n)-n-1)); # _Muniru A Asiru_, Dec 16 2018
%Y Cf. A000010, A000043, A000203, A001229, A018784, A033632, A048050, A132793.
%K hard,more,nonn
%O 1,1
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, Aug 31 2007
%E a(1) corrected and a(6)-a(7) from _Michel Marcus_, Nov 01 2014
%E a(8) from _Giovanni Resta_, Dec 01 2019
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