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%I #8 Jan 10 2019 23:36:47
%S 4769856,23849280,52468416,81087552,90627264,109706688,138325824,
%T 147865536,176484672,195564096,205103808,224183232,252802368,
%U 262342080,281421504,290961216,319580352,338659776,348199488,357739200,376818624,395898048,405437760,424517184
%N List of e-unitary perfect numbers that are not e-semiproper perfect numbers.
%C The e-unitary perfect numbers are numbers k such that the sum of their exponential unitary divisors (A322857) equals 2k. The e-semiproper perfect numbers are numbers k such that the sum of their exponential semiproper divisors (A323309) equals 2k. Apparently most of the e-unitary perfect numbers are also e-semiproper perfect numbers: The first 41393 e-unitary perfect numbers are also the first 41393 e-semiproper perfect numbers, but the 41394th e-unitary perfect number is 4769856 which is not e-semiproper perfect. This number, which is the first term of this sequence, was found by Minculete.
%H Nicusor Minculete, <a href="http://webbut.unitbv.ro/BU2014/Series%20III/BULETIN%20III%20PDF/4.Minculete-MOD.pdf">A new class of divisors: the exponential semiproper divisors</a>, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7 No. 1 (2014), pp. 37-46.
%t fs[p_, e_] := If[e==1, p, p^e + p]; a[1]=1; essigma[n_] := Times @@ fs @@@ FactorInteger[n]; esPerfectQ[n_] := essigma[n]==2n; fu[p_, e_] := DivisorSum[e, p^# &, GCD[#, e/#]==1 &]; eusigma[n_] := Times @@ fu @@@ FactorInteger[n]; euPerfectQ[n_] := eusigma[n] == 2n; aQ[n_] := euPerfectQ[n] && !esPerfectQ[n]; Select[Range[1, 10^8], aQ]
%Y Cf. A322857, A322858, A323308, A323309.
%K nonn
%O 1,1
%A _Amiram Eldar_, Jan 10 2019
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