1,1

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

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]

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.

# 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

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]

(* Alternative: *)

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 *)

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

Sequence in context: A110049 A246577 A097032 * A358617 A384956 A008968

Adjacent sequences: A127658 A127659 A127660 * A127662 A127663 A127664

Ant King, Jan 26 2007

approved