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Numbers k such that phi(sigma(k)) - sigma(phi(k)) = 1.
0

%I #10 Jan 23 2025 08:26:08

%S 2,6,8,24,128,384,32768,98304,2147483648,6442450944

%N Numbers k such that phi(sigma(k)) - sigma(phi(k)) = 1.

%C If k is a natural number less than 6 then 2^(2^k-1) is in the sequence because phi(sigma(2^(2^k-1))) = phi(2^(2^k)-1) = phi((2^(2^0)+1)*(2^(2^1)+1)*...*(2^(2^(k-1))+1)) = 2^(2^0)*2^(2^1)*...*2^(2^(k-1)) = 2^(2^0+2^1+...+2^(k-1)) = 2^(2^k-1) and sigma(phi(2^(2^k-1))) = sigma(2^(2^k-2)) = 2^(2^k-1)-1 so phi(sigma(2^(2^k-1))) - sigma(phi(2^(2^k-1))) = 1 (note that for i = 0,1,2,3 & 4 the Fermat number 2^(2^i)+1 is prime). Next term is greater than 7*10^8.

%C Also if n is a natural number less than 6 then 3*2^(2^n-1) is in the sequence, the proof is similar to the case m=2^(2^n-1). Note that all known terms of the sequence are the ten numbers m*2^(2^n-1) m=1 & 3 and n=1,2,3,4 & 5. Conjecture: There is no other term. - _Farideh Firoozbakht_, Mar 24 2006

%C There are no more terms up to 11*10^9. - _Farideh Firoozbakht_, Apr 13 2010

%F For n<11, a(n)=((-1)^n+2)*2^(2^floor((n+1)/2)-1). - _Farideh Firoozbakht_, Mar 24 2006

%e phi(sigma(2147483648)) = 2147483648 and sigma(phi(2147483648)) = 2147483647 so phi(sigma(2147483648))-sigma(phi(2147483648)) = 1. Hence 2147483648 is in the sequence.

%t Do[If[EulerPhi[DivisorSigma[1,n]]-DivisorSigma[1,EulerPhi[n]]==1, Print[n]],{n,700000000}]

%t Table[((-1)^n + 2)*2^(2^Floor[(n + 1)/2] - 1), {n, 10}] (* _Farideh Firoozbakht_, Mar 24 2006 *)

%o (PARI) is(n)=eulerphi(sigma(n))-sigma(eulerphi(n))==1 \\ _Charles R Greathouse IV_, May 15 2013

%Y Cf. A001229.

%K nonn,more,hard

%O 1,1

%A _Farideh Firoozbakht_, Mar 12 2006