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

Author

Ant King, Jan 25 2007

Keywords

• nonn

Examples

			a(5) = 468 because the fifth integer whose exponential aliquot sequences ends in an e-perfect number is 468.
		

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]

Crossrefs

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

Programs

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 *)