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

17424, 87120, 121968, 226512, 296208, 331056, 400752, 505296, 540144, 609840, 644688, 714384, 749232, 818928, 923472, 1028016, 1062864, 1132560, 1167408, 1237104, 1271952, 1306800, 1376496, 1446192, 1481040, 1550736, 1585584, 1655280, 1690128, 1759824

Offset: 1

Author

Amiram Eldar, Dec 29 2018

Keywords

• nonn

Comments

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.

Minculete and Tóth ask if there is any e-unitary perfect number which is not e-perfect.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Crossrefs

Cf. A051377, A054979, A054980, A322857.

Programs

Mathematica

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]