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A322858 List of e-perfect numbers that are not e-unitary perfect.

%I A322858 #23 May 30 2025 03:50:31
%S A322858 17424,87120,121968,226512,296208,331056,400752,505296,540144,609840,
%T A322858 644688,714384,749232,818928,923472,1028016,1062864,1132560,1167408,
%U A322858 1237104,1271952,1306800,1376496,1446192,1481040,1550736,1585584,1655280,1690128,1759824
%N A322858 List of e-perfect numbers that are not e-unitary perfect.
%C A322858 The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. Apparently most of the e-perfect numbers (A054979) are also e-unitary perfect numbers: the first 150 e-perfect numbers are also the first 150 e-unitary perfect numbers. But A054979(151) = 17424 is not e-unitary perfect.
%C A322858 Minculete and Tóth ask if there is any e-unitary perfect number which is not e-perfect.
%C A322858 The asymptotic density of this sequence is Sum_{n>=1} f(b(n)) = 0.000016169..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)) and b = {17424, 1306800, 54531590400, ...} is the sequence of primitive e-perfect numbers (A054980) that are not e-unitary perfect. - _Amiram Eldar_, May 06 2025
%H A322858 Amiram Eldar, <a href="/A322858/b322858.txt">Table of n, a(n) for n = 1..10000</a>
%H A322858 Nicuşor Minculete and László Tóth, <a href="https://doi.org/10.71352/ac.35.205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.
%t A322858 f[p_, e_] := DivisorSum[e, p^# &]; esigma[n_] := Times @@ f @@@ FactorInteger[n]; ePerfectQ[n_] := esigma[n] == 2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := ePerfectQ[n] && !euPerfectQ[n]; Select[Range[125000], aQ]
%Y A322858 Cf. A051377, A054979, A054980, A322857.
%K A322858 nonn
%O A322858 1,1
%A A322858 _Amiram Eldar_, Dec 29 2018