A127658 Exponential aspiring numbers.

900, 1352, 1728, 2880, 2916, 3000, 3750, 4356, 5292, 6480, 6760, 8100, 8640, 9464, 9900, 10404, 10648, 11700, 12000, 12096, 13500, 14580, 14872, 15300, 15552, 15876, 16000, 16200, 16224, 17100, 17836, 18252, 19008, 19044, 20160, 20412, 20700, 21780, 22464, 22500

Offset

1,1

Comments

Exponential aspiring numbers are those integers whose exponential aliquot sequences end in an e-perfect number, but that are not e-perfect numbers themselves.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Peter Hagis, Jr., Some results concerning exponential divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]

J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]

J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Example

a(5) = 2916 because the fifth non-e-perfect number whose exponential aliquot sequence ends in an e-perfect number is 2916.

Mathematica

ExponentialDivisors[1]={1}; ExponentialDivisors[n_]:=Module[{}, {pr, pows}=Transpose@FactorInteger[n]; divpowers=Distribute[Divisors[pows], List]; Sort[Times@@(pr^Transpose[divpowers])]]; se[n_]:=Plus@@ExponentialDivisors[n]-n; g[n_] := If[n > 0, se[n], 0]; eTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[25000], ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] && !ExponentialPerfectNumberQ[ # ]&]
f[p_, e_] := DivisorSum[e, p^# &]; s[0] = s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, v[[-1]] != n && v[[-2]] == v[[-1]] > 0]; Select[Range[32000], q] (* Amiram Eldar, Mar 11 2023 *)

Crossrefs

Cf. A127656, A127657, A127659, A127660, A054979.

Sequence in context: A061044 A344595 A344694 * A318720 A338540 A137490

Adjacent sequences: A127655 A127656 A127657 * A127659 A127660 A127661

Keyword

nonn

Author

Ant King, Jan 25 2007

Status

approved