A127656 Lengths of the exponential aliquot sequences.

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 3, 4, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4

Offset

1,1

Comments

The exponential aliquot sequence is defined by the map x -> A051377(x)-x starting at n.

The length of an exponential aliquot sequence is defined according to the length of its transient part + the length of its terminal cycle.

Links

Hans Havermann, Table of n, a(n) for n = 1..10000

Hagis, Peter 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(4)=3 because the exponential aliquot sequence generated by 4 is <4,2,0> and it has length 3.
From R. J. Mathar, Oct 05 2017: (Start)
The aliquot sequnence may enter a cycle (see A054979)
36 -> 36 -> ..
180 -> 180 -> ..
252 -> 252 -> ..
396 -> 396 -> ..
468 -> 468 -> ..
612 -> 612 -> ..
684 -> 684 -> ..
828 -> 828 -> ..
900 -> 1260 -> 1260 -> ..
1044 -> 1044 -> ..
1116 -> 1116 -> ..
1260 -> 1260 -> ..
1332 -> 1332 -> ..
1352 -> 468 -> 468 -> ..
1476 -> 1476 -> ..
1548 -> 1548 -> ..
1692 -> 1692 -> ..
1728 -> 612 -> 612 -> ..
1800 -> 1800 -> ..
1908 -> 1908 -> ..
1980 -> 1980 -> ..
2124 -> 2124 -> ..
2196 -> 2196 -> ..
2340 -> 2340 -> ..
2412 -> 2412 -> ..
2556 -> 2556 -> ..
2628 -> 2628 -> ..
2700 -> 2700 -> ..
2772 -> 2772 -> ..
2844 -> 2844 -> ..
2880 -> 1800 -> 1800 -> ..
(End)

Maple

A127656 := proc(n)
    local trac, x;
    x := n ;
    trac := [x] ;
    while true do
        x := A051377(x)-trac[-1] ;
        if x = 0 then
            return 1+nops(trac) ;
        elif x in trac then
            return nops(trac) ;
        end if;
        trac := [op(trac), x] ;
    end do:
end proc: # R. J. Mathar, Oct 05 2017

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]]; Length[eTrajectory[ # ]] &/@Range[100]
(* Second program: *)
f[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}]&) /@ FactorInteger[n];
a[n_] := Length[FixedPointList[f[#]-#&, n]]-1;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 04 2023 *)

Crossrefs

Cf. A127657, A127658, A127659, A127660.

Cf. also A127661.

Sequence in context: A087040 A065569 A262941 * A109709 A125604 A216685

Adjacent sequences: A127653 A127654 A127655 * A127657 A127658 A127659

Keyword

nonn

Author

Ant King, Jan 25 2007

Status

approved