A127657 Integers whose exponential aliquot sequences end in an e-perfect number.

36, 180, 252, 396, 468, 612, 684, 828, 900, 1044, 1116, 1260, 1332, 1352, 1476, 1548, 1692, 1728, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2880, 2916, 2988, 3000, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3750, 3852, 3924, 4068, 4140

Offset

1,1

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) = 468 because the fifth integer whose exponential aliquot sequences ends in an e-perfect number is 468.

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]]; ExponentialPerfectNumberQ[0]=False; ExponentialPerfectNumberQ[k_Integer] :=If[se[k]==k, True, False]; Select[Range[5000], ExponentialPerfectNumberQ[Last[eTrajectory[ # ]]] &]
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]] == v[[-2]] > 0]; Select[Range[4000], q] (* Amiram Eldar, Mar 11 2023 *)

Crossrefs

Cf. A127656, A127658, A127659, A127660, A054979.

Sequence in context: A219858 A307958 A348962 * A318100 A335218 A321145

Adjacent sequences: A127654 A127655 A127656 * A127658 A127659 A127660

Keyword

nonn

Author

Ant King, Jan 25 2007

Status

approved