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A127656 Lengths of the exponential aliquot sequences.

%I A127656 #18 Jun 04 2023 18:57:17
%S A127656 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,
%T A127656 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,
%U A127656 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
%N A127656 Lengths of the exponential aliquot sequences.
%C A127656 The exponential aliquot sequence is defined by the map x -> A051377(x)-x starting at n.
%C A127656 The length of an exponential aliquot sequence is defined according to the length of its transient part + the length of its terminal cycle.
%H A127656 Hans Havermann, <a href="/A127656/b127656.txt">Table of n, a(n) for n = 1..10000</a>
%H A127656 Hagis, Peter Jr., <a href="http://dx.doi.org/10.1155/S0161171288000407">Some Results Concerning Exponential Divisors</a>, Internat. J. Math. & Math. Sci., Vol. 11, No. 2, (1988), pp. 343-350.
%H A127656 J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]
%H A127656 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 A127656 J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]
%e A127656 a(4)=3 because the exponential aliquot sequence generated by 4 is <4,2,0> and it has length 3.
%e A127656 From _R. J. Mathar_, Oct 05 2017: (Start)
%e A127656 The aliquot sequnence may enter a cycle (see A054979)
%e A127656 36 -> 36 -> ..
%e A127656 180 -> 180 -> ..
%e A127656 252 -> 252 -> ..
%e A127656 396 -> 396 -> ..
%e A127656 468 -> 468 -> ..
%e A127656 612 -> 612 -> ..
%e A127656 684 -> 684 -> ..
%e A127656 828 -> 828 -> ..
%e A127656 900 -> 1260 -> 1260 -> ..
%e A127656 1044 -> 1044 -> ..
%e A127656 1116 -> 1116 -> ..
%e A127656 1260 -> 1260 -> ..
%e A127656 1332 -> 1332 -> ..
%e A127656 1352 -> 468 -> 468 -> ..
%e A127656 1476 -> 1476 -> ..
%e A127656 1548 -> 1548 -> ..
%e A127656 1692 -> 1692 -> ..
%e A127656 1728 -> 612 -> 612 -> ..
%e A127656 1800 -> 1800 -> ..
%e A127656 1908 -> 1908 -> ..
%e A127656 1980 -> 1980 -> ..
%e A127656 2124 -> 2124 -> ..
%e A127656 2196 -> 2196 -> ..
%e A127656 2340 -> 2340 -> ..
%e A127656 2412 -> 2412 -> ..
%e A127656 2556 -> 2556 -> ..
%e A127656 2628 -> 2628 -> ..
%e A127656 2700 -> 2700 -> ..
%e A127656 2772 -> 2772 -> ..
%e A127656 2844 -> 2844 -> ..
%e A127656 2880 -> 1800 -> 1800 -> ..
%e A127656 (End)
%p A127656 A127656 := proc(n)
%p A127656     local trac,x;
%p A127656     x := n ;
%p A127656     trac := [x] ;
%p A127656     while true do
%p A127656         x := A051377(x)-trac[-1] ;
%p A127656         if x = 0 then
%p A127656             return 1+nops(trac) ;
%p A127656         elif x in trac then
%p A127656             return nops(trac) ;
%p A127656         end if;
%p A127656         trac := [op(trac),x] ;
%p A127656     end do:
%p A127656 end proc: # _R. J. Mathar_, Oct 05 2017
%t A127656 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]
%t A127656 (* Second program: *)
%t A127656 f[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}]&) /@ FactorInteger[n];
%t A127656 a[n_] := Length[FixedPointList[f[#]-#&, n]]-1;
%t A127656 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 04 2023 *)
%Y A127656 Cf. A127657, A127658, A127659, A127660.
%Y A127656 Cf. also A127661.
%K A127656 nonn
%O A127656 1,1
%A A127656 _Ant King_, Jan 25 2007