A127657 - OEIS

%I #13 Mar 11 2023 08:05:01

%S 36,180,252,396,468,612,684,828,900,1044,1116,1260,1332,1352,1476,

%T 1548,1692,1728,1800,1908,1980,2124,2196,2340,2412,2556,2628,2700,

%U 2772,2844,2880,2916,2988,3000,3060,3204,3276,3420,3492,3636,3708,3750,3852,3924,4068,4140

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

%H Amiram Eldar, <a href="/A127657/b127657.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Hagis, Jr., <a href="https://doi.org/10.1155/S0161171288000407">Some results concerning exponential divisors</a>, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.

%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Broken link]

%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>. [Via Internet Archive Wayback-Machine]

%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]

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

%t 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[ # ]]] &]

%t 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 *)

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

%K nonn

%O 1,1

%A _Ant King_, Jan 25 2007