A127659 Exponential amicable numbers.

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.

The asymptotic density of this sequence is Sum_{n>=1} (f(A323753(n)) + f(A323754(n))) = 5.94917...*10^(-6), where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Dec 05 2025

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Peter Hagis, Jr., Some results concerning exponential divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-349.

J. O. M. Pedersen, Tables of Aliquot Cycles.

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 *)

Crossrefs

Cf. A126164, A126165, A126166, A127656, A127657, A127658, A127660, A051377, A323753, A323754.

Sequence in context: A015316 A212853 A127660 * A126165 A323753 A255985

Adjacent sequences: A127656 A127657 A127658 * A127660 A127661 A127662

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