A127661 Lengths of the infinitary aliquot sequences.

2, 3, 3, 3, 3, 1, 3, 4, 3, 5, 3, 5, 3, 6, 4, 3, 3, 6, 3, 6, 4, 7, 3, 8, 3, 4, 4, 6, 3, 6, 3, 4, 5, 7, 4, 7, 3, 8, 4, 8, 3, 5, 3, 4, 5, 5, 3, 7, 3, 7, 5, 7, 3, 4, 4, 6, 4, 5, 3, 1, 3, 8, 4, 5, 4, 3, 3, 8, 5, 10, 3, 3, 3, 9, 4, 9, 4, 2, 3, 8, 3, 5, 3, 10, 4, 6, 6, 8, 3, 1, 5, 7, 5, 8, 4, 9, 3, 8, 5, 7

Offset: 1

Author

Ant King, Jan 26 2007

Keywords

• nonn

Comments

An infinitary aliquot sequence is defined by the map x->A049417(x)-x. The map usually terminates with a zero, but may enter cycles (if n in A127662 for example).

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

The value of a(840) starting the infinitary aliquot sequence 840 -> 2040 -> 4440 -> 9240 -> 25320,... is >1500. - R. J. Mathar, Oct 05 2017

Examples

			a(4)=3 because the infinitary aliquot sequence generated by 4 is 4 -> 1 -> 0 and it has length 3.
a(6) = 1 because 6 -> 6 -> 6 ->... enters a cycle after 1 term.
a(8) = 4 because 8 -> 7 -> 1 -> 0 terminates after 4 terms.
a(30) = 6 because 30 ->42 -> 54 -> 66 -> 78 -> 90 -> 90 -> 90 -> ...enters a cycle after 6 terms.
a(126)=2 because 126 -> 114 -> 126 enters a cycle after 2 terms.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..839
• Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
• Hans Havermann, Graphs of infinitary aliquot sequences for 840, 1152, 2442, 2658, 2982, 5766, 6216, 6870, 7560, 8670, 9030, 9570 (click to see full plots)
• D. Moews, A database of aliquot cycles (2015)
• 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]

Crossrefs

Cf. A126168, A127662 - A127667, A293355, A098007.

Programs

Maple

# Uses code snippets of A049417
A127661 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A049417(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:
seq(A127661(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

Mathematica

ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ]; properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Length[iTrajectory[ # ]] &/@ Range[100]
(* Second program: *)
A049417[n_] := If[n == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
A127661[n_] := Module[{trac, x}, x = n; trac = {x}; While[True, x = A049417[x] - trac[[-1]]; If[x == 0, Return[1 + Length[trac]], If[MemberQ[trac, x], Return[Length[trac]]]]; trac = Append[trac, x]]];
Table[A127661[n], {n, 1, 100}] (* Jean-François Alcover, Aug 28 2023, after R. J. Mathar *)