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
KEYWORD
nonn
AUTHOR
Ant King, Jan 25 2007
STATUS
approved