This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A323310 #17 May 06 2025 09:31:38
%S A323310 4769856,23849280,52468416,81087552,90627264,109706688,138325824,
%T A323310 147865536,176484672,195564096,205103808,224183232,252802368,
%U A323310 262342080,281421504,290961216,319580352,338659776,348199488,357739200,376818624,395898048,405437760,424517184
%N A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.
%C A323310 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.
%C A323310 The powerful (A001694) terms of this sequence are the primitive terms, i.e., if k is a powerful term, then m*k is a term for any squarefree (A005117) number m that is coprime to k. The only primitive terms below 10^18 are 4769856 and 357739200. If S is the sequence of primitive terms, then the asymptotic density of this sequence is Sum_{n>=1} f(S(n)) = 5.235...*10^(-8), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - _Amiram Eldar_, May 06 2025
%H A323310 Amiram Eldar, <a href="/A323310/b323310.txt">Table of n, a(n) for n = 1..10000</a>
%H A323310 Nicusor Minculete, <a href="https://webbut.unitbv.ro/index.php/Series_III/article/view/1943">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 A323310 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 A323310 Cf. A322857, A322858, A323308, A323309.
%K A323310 nonn
%O A323310 1,1
%A A323310 _Amiram Eldar_, Jan 10 2019