OFFSET
1,1
COMMENTS
The aliquot sequence for n is the trajectory of n under repeated application of the map x -> sigma(x) - x (= A001065).
The trajectory will either have a transient part followed by a cyclic part, or will have an infinite transient part and never cycle. It seems possible that this be the case for 276, i.e., a(276) = -1.
Sequence gives number of distinct terms in the trajectory = (length of transient part of trajectory) + (length of cycle (which is 1 if the trajectory reached 0)), or -1 if the sequence never cycles.
Concerning one of the previously unsolved cases, Robert G. Wilson v reports that 840 reaches 0 after 749 iterations. - Sep 10 2004
Up to 1000 there are 12 numbers whose fate is currently unknown, namely five well-known hard cases: 276, 552, 564, 660, 966 and seven others: 306, 396 and 696, all on same trajectory as 276; 780, on same trajectory as 564; 828, on same trajectory as 660; 888, on same trajectory as 552; 996, on same trajectory as 660. - T. D. Noe, Jun 06 2006
The sum-of-divisors function sigma (A000203) and thus aliquot parts A001065 are defined only on the positive integers, so the trajectory ends when 0 is reached. Some authors define A001065 to be the sum of the positive numbers less than n that divide n, in which case one would have A001065(0) = 0. - M. F. Hasler, Nov 16 2013
REFERENCES
K. Chum, R. K. Guy, M. J. Jacobson, Jr., and A. S. Mosunov, Numerical and statistical analysis of aliquot sequences. Exper. Math. 29 (2020), no. 4, 414-425; arXiv:2110.14136, Oct. 2021 [math.NT].
J.-P. Delahaye, Les inattendus mathématiques, Chapter 19, "Nombres amiables et suites aliquotes", pp. 217-229, Belin-Pour la Science, Paris 2004.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B6.
R. K. Guy and J. L. Selfridge, Interim report on aliquot series, pp. 557-580 of Proceedings Manitoba Conference on Numerical Mathematics. University of Manitoba, Winnipeg, Oct 1971.
Carl Pomerance, The aliquot constant, after Bosma and Kane, Q. J. Math. 69 (2018), no. 3, 915-930.
LINKS
T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..275
Christophe Clavier, Aliquot Sequences
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..1000 with "NA" for unknown terms
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 14.
Juan L. Varona, List of "primitive" numbers not known to terminate (Oct 19 2004: list begins 276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, ...) [This is not the full list of numbers not known to terminate - see Comments above]
Eric Weisstein's World of Mathematics, Aliquot Sequence.
Wikipedia, Aliquot sequence
P. Zimmermann, Aliquot Sequences
EXAMPLE
Examples of trajectories:
1, 0.
2, 1, 0.
3, 1, 0. (and similarly for any prime)
4, 3, 1, 0.
5, 1, 0.
6, 6, 6, ... (and similarly for any perfect number)
8, 7, 1, 0.
9, 4, 3, 1, 0.
12, 16, 15, 9, 4, 3, 1, 0.
14, 10, 8, 7, 1, 0.
25, 6, 6, 6, ...
28, 28, 28, ... (the next perfect number)
30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0.
42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0.
MAPLE
f:=proc(n) local t1, i, j, k; t1:=[n]; for i from 2 to 50 do j:= t1[i-1]; k:=sigma(j)-j; t1:=[op(t1), k]; od: t1; end; # produces trajectory for n
# 2nd implementation:
A098007 := proc(n)
local trac, x;
x := n ;
trac := [x] ;
while true do
x := numtheory[sigma](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(A098007(n), n=1..100) ; # R. J. Mathar, Oct 08 2017
MATHEMATICA
g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 1, {n, 100}] (* Robert G. Wilson v, Sep 10 2004 *)
PROG
(Scheme)
(define (A098007 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) (+ 1 i)) ((member next visited) i) (else (loop (cons next visited) (+ 1 i)))))))
;; Antti Karttunen, Nov 01 2017
(PARI) apply( {A098007(n, t=0)=until(bittest(t, if(n, n=sigma(n)-n)), t+=1<<n); hammingweight(t)}, [1..99]) \\ M. F. Hasler, Feb 24 2018, improved Aug 14 2022 thanks to a remark from Jianing Song
(Python)
from sympy import divisor_sigma as sigma
def a(n, limit=float('inf')):
alst = []; seen = set(); i = n; c = 0
while i and i not in seen and c < limit:
alst.append(i); seen.add(i); i = sigma(i) - i; c += 1
return "NA" if c == limit else len(set(alst + [i]))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 11 2021
CROSSREFS
Cf. A001065.
There are many related sequences:
Length of transient part + length of cycle: this sequence. Other versions of the current sequence: A044050, A003023.
Numbers which eventually reach 1 (or equivalently 0): A080907.
Aliquot trajectories for certain interesting starting values: A008885 (for 30), A008886 A008887 A008888 A008889 A008890 A008891 A008892 (for 276), A014360 A014361 A074907 A014362 A045477 A014363 A014364 A014365 A074906, A171103.
For n < 220, A098008 = A098007 - 1, i.e., 220 is the first sociable number. - Robert G. Wilson v, Sep 10 2004
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Sep 09 2004
EXTENSIONS
More terms from Robert G. Wilson v and John W. Layman, Sep 10 2004
STATUS
approved