OFFSET
1,1
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
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]
EXAMPLE
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.
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;
CROSSREFS
KEYWORD
nonn
AUTHOR
Ant King, Jan 26 2007
STATUS
approved