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A127659
Exponential amicable numbers.
5
90972, 100548, 454860, 502740, 937692, 968436, 1000692, 1106028, 1182636, 1307124, 1546524, 1709316, 2092356, 2312604, 2638188, 2820132, 2915892, 3116988, 3365964, 3720276, 3729852, 3911796, 4122468, 4275684, 4323564, 4548600, 4688460, 4725756, 4821516, 4842180
OFFSET
1,1
COMMENTS
Union of A126165 and A126166. The first 10 terms of this sequence are the same as the first 10 terms of A127660.
REFERENCES
Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
LINKS
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]
FORMULA
Non-e-perfect numbers for which A126164(A126164(n))=n.
EXAMPLE
a(5)=937692 because the fifth non-e-perfect integer that satisfies A126164(A126164(n))=n is 937692.
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[ # ] &]
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 *)
KEYWORD
nonn
AUTHOR
Ant King, Jan 25 2007
EXTENSIONS
Link corrected by Andrew Lelechenko, Dec 04 2011
More terms from Amiram Eldar, May 09 2019
STATUS
approved