A323310 List of e-unitary perfect numbers that are not e-semiproper perfect numbers.

4769856, 23849280, 52468416, 81087552, 90627264, 109706688, 138325824, 147865536, 176484672, 195564096, 205103808, 224183232, 252802368, 262342080, 281421504, 290961216, 319580352, 338659776, 348199488, 357739200, 376818624, 395898048, 405437760, 424517184

Offset: 1

Author

Amiram Eldar, Jan 10 2019

Keywords

• nonn

Comments

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.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nicusor Minculete, A new class of divisors: the exponential semiproper divisors, Bulletin of the Transilvania University of Brasov, Mathematics, Informatics, Physics, Series III, Vol. 7, No. 1 (2014), pp. 37-46.

Crossrefs

Cf. A322857, A322858, A323308, A323309.

Programs

Mathematica

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]