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A120116 Numbers n such that sigma(uphi(n)) = n where uphi is the unitary totient (or unitary phi) function (see A047994). 1


%S 1,3,4,8,12,15,24,32,60,96,120,128,255,384,480,1020,1920,2040,2418,

%T 8160,8192,24576,32640,65535,122880,131072,262140,370986,393216,

%U 524280,524288,1572864,1966080,2088960,2097120,7864320,8388480,33423360,110771178,133693440

%N Numbers n such that sigma(uphi(n)) = n where uphi is the unitary totient (or unitary phi) function (see A047994).

%C Theorem: If 2^p-1 is prime (a Mersenne prime) and 0<=k<=5 then m=2^p*(2^2^k-1) is in the sequence. Proof: If k=0 then m=2^p and sigma(uphi(m))=sigma(uphi(2^p))=sigma(2^p-1)=2^p=m. Now if k>0 then m=2^p*(2^2^0+1)*(2^2^1+1)*...*(2^2^(k-1)+1) since 2^2^i+1 for i=0,1,...,4 are primes(Fermat primes) we conclude that uphi(m)=(2^p-1)*(2^2^0)*(2^2^1)*...*(2^2^(k-1))=(2^p-1)*2^ (2^0+2^1+...+2^(k-1))=(2^p-1)*2^(2^k-1) hence sigma(uphi(m))= sigma(2^p-1)*sigma(2^(2^k-1))=2^p*(2^2^k-1)=m and the proof is complete. 2418, 370986 & 110771178 are the only terms up to 15*10^8 which aren't of the form 2^p*(2^2^k-1).

%H Donovan Johnson, <a href="/A120116/b120116.txt">Table of n, a(n) for n = 1..50</a> (terms < 2^35)

%e 110771178 = 2*3*7*19*127*1093 is in the sequence because sigma(uphi(2*3*7*19*127*1093)) = sigma((2-1)*(3-1)*(7-1)*(19-1)*(127-1)*(1093-1) = sigma(2*6*18*126*1092) = sigma(2^6*3^6*7^2*13) = (2^7-1)*((3^7-1)/2)*((7^3-1)/6)*14 = 110771178.

%t uphi[n_] := (A = FactorInteger[n]; l = Length[A]; Product[A[[k]][[1]] ^A[[k]][[2]] - 1, {k, l}]); Do[If[DivisorSigma[1, uphi[n]] == n, Print[n]], {n, 150000000}]

%Y Cf. A047994, A019434.

%K nonn

%O 1,2

%A _Farideh Firoozbakht_, Jul 11 2006

%E Name corrected by _Donovan Johnson_, May 01 2013

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Last modified January 22 14:05 EST 2021. Contains 340362 sequences. (Running on oeis4.)