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%I #18 May 09 2019 15:41:46
%S 90972,100548,454860,502740,937692,968436,1000692,1106028,1182636,
%T 1307124,1546524,1709316,2092356,2312604,2638188,2820132,2915892,
%U 3116988,3365964,3720276,3729852,3911796,4122468,4275684,4323564,4548600,4688460,4725756,4821516,4842180
%N Exponential amicable numbers.
%C Union of A126165 and A126166. The first 10 terms of this sequence are the same as the first 10 terms of A127660.
%D Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
%H Amiram Eldar, <a href="/A127659/b127659.txt">Table of n, a(n) for n = 1..10000</a>
%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]
%F Non-e-perfect numbers for which A126164(A126164(n))=n.
%e a(5)=937692 because the fifth non-e-perfect integer that satisfies A126164(A126164(n))=n is 937692.
%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]];ExponentialAmicableNumberQ[k_]:=If[Nest[se,k,2]==k && !se[k]==k,True,False];Select[Range[5 10^6],ExponentialAmicableNumberQ[ # ] &]
%t fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; s = {}; Do[m = esigma[n] - n; If[m != n && esigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^6}]; s (* _Amiram Eldar_, May 09 2019 *)
%Y Cf. A126164, A126165, A126166, A127656, A127657, A127658, A127660, A051377.
%K nonn
%O 1,1
%A _Ant King_, Jan 25 2007
%E Link corrected by _Andrew Lelechenko_, Dec 04 2011
%E More terms from _Amiram Eldar_, May 09 2019
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