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

1,1

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.

Links

Table of n, a(n) for n=1..24.

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.

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]

Crossrefs

Cf. A322857, A322858, A323308, A323309.

Sequence in context: A258132 A083641 A224641 * A017083 A017167 A017383

Adjacent sequences: A323307 A323308 A323309 * A323311 A323312 A323313

Keyword

nonn

Author

Amiram Eldar, Jan 10 2019

Status

approved