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A098007 Length of aliquot sequence for n, or -1 if aliquot sequence never cycles. 36
2, 3, 3, 4, 3, 1, 3, 4, 5, 5, 3, 8, 3, 6, 6, 7, 3, 5, 3, 8, 4, 7, 3, 6, 2, 8, 4, 1, 3, 16, 3, 4, 7, 9, 4, 5, 3, 8, 4, 5, 3, 15, 3, 6, 8, 9, 3, 7, 5, 4, 5, 10, 3, 14, 4, 6, 4, 5, 3, 12, 3, 10, 4, 5, 4, 13, 3, 6, 5, 7, 3, 10, 3, 6, 6, 6, 4, 12, 3, 8, 6, 7, 3, 7, 4, 10, 8, 8, 3, 11, 5, 7, 5, 5, 3, 10, 3, 4, 5, 6 (list; graph; refs; listen; history; text; internal format)
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.

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.

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

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.

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

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

Index entries for sequences related to aliquot parts.

EXAMPLE

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

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)))))))

(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.

;; Antti Karttunen, Nov 01 2017

(PARI) A098007(n)=for(k=1, oo, ((n!=n=sigma(n)-n)&&n)||return(k+!n)) \\ M. F. Hasler, Feb 24 2018

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.

Length of transient part: A098008, also A007906. Records for transients: A098009, A098010.

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

Sequence in context: A159909 A176004 A214738 * A278116 A215469 A007554

Adjacent sequences:  A098004 A098005 A098006 * A098008 A098009 A098010

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

Concerning one of the previously unsolved cases, Robert G. Wilson v reports that 840 reaches 0 after 749 iterations. - Sep 10 2004

STATUS

approved

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Last modified June 26 10:12 EDT 2019. Contains 324375 sequences. (Running on oeis4.)