OFFSET
1,1
COMMENTS
Sometimes called the exponential 2-cycle attractor set. The first 10 terms of this sequence are the same as the first 10 terms of A127659.
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(11) = 1383732 because the eleventh integer whose exponential aliquot sequence ends in an exponential amicable pair is 1383732.
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]]; ExponentialAmicableNumberQ[k_]:=If[Nest[se, k, 2]==k && !se[k]==k, True, False]; Select[Range[5 10^6], ExponentialAmicableNumberQ[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[[-2]] != v[[-1]] > 0 && v[[-3]] == v[[-1]]]; Select[Range[10^6], q] (* Amiram Eldar, Mar 11 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ant King, Jan 25 2007
STATUS
approved