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The EulerPhi(sigma(EulerPhi))-perfect numbers, where the f-perfect numbers for an arithmetical function f are defined in A066218.
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%I #15 Aug 18 2024 19:02:56

%S 2,4,28,40,448,500380

%N The EulerPhi(sigma(EulerPhi))-perfect numbers, where the f-perfect numbers for an arithmetical function f are defined in A066218.

%C There are no terms between 449 and 10^5. Are there any more terms? Are there infinitely many?

%C No more terms below 10^9. - _Amiram Eldar_, Sep 26 2019

%H J. Pe, <a href="http://jlpe.tripod.com/gpn/fperfect.htm">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.

%e Let f(n) = EulerPhi(sigma(EulerPhi(n))). Proper divisors of 28 = {1, 2, 4, 7, 14}; the sum of their f-values = 1+1+2+4+4 = 12 = f(28); hence 28 belongs to the sequence.

%t f[x_] := EulerPhi[DivisorSigma[1, EulerPhi[x]]]; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

%Y Cf. A000010, A000203, A066218.

%K nonn,more

%O 1,1

%A _Joseph L. Pe_, Dec 18 2001

%E a(6) from _Amiram Eldar_, Sep 26 2019