A044050 -id:A044050 - cp's OEIS frontend

A098007 Length of aliquot sequence for n, or -1 if aliquot sequence never cycles.

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

Offset: 1

N. J. A. Sloane, Sep 09 2004

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

			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.

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.

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
Index entries for sequences related to aliquot parts.

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

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

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 *)

apply( {A098007(n, t=0)=until(bittest(t,if(n,n=sigma(n)-n)),t+=1<M. F. Hasler, Feb 24 2018, improved Aug 14 2022 thanks to a remark from Jianing Song

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

(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

More terms from Robert G. Wilson v and John W. Layman, Sep 10 2004

A098008 Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

Original entry on oeis.org

1, 2, 2, 3, 2, 0, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, 1, 7, 3, 0, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 13, 3, 5, 3, 4, 2, 11, 2, 9, 3, 4, 3, 12, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, 11, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, 10, 4, 6, 4, 4, 2, 9, 2, 3, 4, 5, 2, 18

Offset: 1

N. J. A. Sloane, Sep 09 2004

See A098007 for further information.
a(n) = 0 if and only if n is perfect (A000396) or part of a cycle of length greater than 1. - Comment corrected by Antti Karttunen, Nov 02 2017.
It is believed that the first time a(n) = -1 is at n = 276 (see A008892). - N. J. A. Sloane, Nov 02 2017

			From _Antti Karttunen_, Nov 02 2017: (Start)
For n = 3, a(n) = 2, because A001065(3) = 1 and A001065(1) = 0, so it took two steps to end in zero.
For n = 25, a(n) = 1, because A001065(25) = 6, and A001065(6) = 6, so it took one step to enter into a cycle.
For n = 12496, a(n) = 0, because 12496 is a member of 5-cycle of map n -> A001065(n) (see A072891).
(End)

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.

Antti Karttunen, Table of n, a(n) for n = 1..275

Cf. A001065, A098007, A044050, A003023, A008892. See A007906 for another version.

Cf. A206708 (gives a proper subset of zeros).

g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; Table[ Length[ f[n]] - 2, {n, 102}] (* good only for n<220 *) (* Robert G. Wilson v, Sep 10 2004 *)

(define (A098008 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) i) ((member next visited) => (lambda (transientplus1) (- (length transientplus1) 1))) (else (loop (cons next visited) (+ 1 i))))))) ;; Good for at least n = 1..275.
(define (A001065 n) (- (A000203 n) n)) ;; For an implementation of A000203, see under that entry.
;; Antti Karttunen, Nov 02 2017

More terms from Robert G. Wilson v, Sep 10 2004

A115350 Termination of the aliquot sequence starting at n.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 3, 7, 11, 3, 13, 7, 3, 3, 17, 11, 19, 7, 11, 7, 23, 17, 6, 3, 13, 28, 29, 3, 31, 31, 3, 7, 13, 17, 37, 7, 17, 43, 41, 3, 43, 43, 3, 3, 47, 41, 7, 43, 11, 3, 53, 3, 17, 41, 23, 31, 59, 43, 61, 7, 41, 41, 19, 3, 67, 31, 13, 43, 71, 3, 73, 43, 7, 41, 19, 3, 79, 41, 43, 43

Offset: 1

Sergio Pimentel, Mar 07 2006

nonn

Catalan's conjecture [not yet established and probably false] is that every aliquot sequence terminates in a prime number followed by 1, a perfect number, a friendly pair or an aliquot cycle.

a(n) = the prime number if the sequence terminates in a prime followed by 1, a(n) = a perfect number if the sequence terminates in a perfect number, a(n) = the smallest number of the cycle if the sequence terminates in an aliquot cycle, a(n) = 0 if the sequence is open ended. So far 276 is the smallest number for which the termination of the aliquot sequence is not known.

			a(12)=3 since the aliquot sequence starting at 12 is: 12 - 16 - 15 - 9 - 4 - 3.
a(95)=6 since the aliquot sequence starting at 95 is: 95 - 25 - 6 - 6 ...

W. Creyaufmueller, Aliquot Sequences.
R. J. Mathar, Table of n, a(n) for n= 1,...,12572 with -1 substituted for a(n) where terminations are not yet known.
R. J. Mathar, Illustration of Aliquot Sequence Mergers (2014)
Paul Zimmerman, Aliquot Sequences.

Cf. A098007, A098008, A098009, A098010, A003023, A044050, A007906, A037020, A063769, A005114.

a[n_] := If[n == 1, 1, FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, n] /. {k__, 1, 0, 0} :> {k} // Last];
Array[a, 100] (* Jean-François Alcover, Mar 28 2020 *)

Edited by N. J. A. Sloane, Aug 14 2006

a(61)-a(80) from R. J. Mathar's list by Robert Price, Mar 16 2019

A007906 Number of steps for aliquot sequence for n to converge to 0, or -1 if it never reaches 0.

Original entry on oeis.org

1, 2, 2, 3, 2, -1, 2, 3, 4, 4, 2, 7, 2, 5, 5, 6, 2, 4, 2, 7, 3, 6, 2, 5, -1, 7, 3, -1, 2, 15, 2, 3, 6, 8, 3, 4, 2, 7, 3, 4, 2, 14, 2, 5, 7, 8, 2, 6, 4, 3, 4, 9, 2, 13, 3, 5, 3, 4, 2, 11, 2, 9, 3, 4, 3, 12, 2, 5, 4, 6, 2, 9, 2, 5, 5, 5, 3, 11, 2, 7, 5, 6, 2, 6, 3, 9, 7, 7, 2, 10, 4, 6, 4, 4, -1, 9, 2, 3

Offset: 1

Michael Gerenrot (sch116(AT)yahoo.com)

sign

Length of transient part of trajectory of n if trajectory reaches 1, otherwise a(n) = -1. See A098008 for another version. See A098007 for further information.

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.

Antti Karttunen, Table of n, a(n) for n = 1..275

Cf. A098008, A098007, A044050, A003023.

(define (A007906 n) (let loop ((visited (list n)) (i 1)) (let ((next (A001065 (car visited)))) (cond ((zero? next) i) ((member next visited) -1) (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 02 2017

Definition changed by N. J. A. Sloane, Nov 02 2017 at the suggestion of Antti Karttunen.

A115060 Maximum peak of aliquot sequence starting at n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 21, 19, 22, 21, 22, 23, 55, 25, 26, 27, 28, 29, 259, 31, 32, 33, 34, 35, 55, 37, 38, 39, 50, 41, 259, 43, 50, 45, 46, 47, 76, 49, 50, 51, 52, 53, 259, 55, 64, 57, 58, 59, 172, 61, 62, 63, 64, 65, 259

Offset: 1

Sergio Pimentel, Mar 06 2006

nonn

According to Catalan's conjecture all aliquot sequences end in a prime followed by 1, a perfect number, a friendly pair or an aliquot cycle. Some sequences seem to be open ended and keep growing forever i.e. 276. Most sequences only go down (i.e. 10 - 8 - 7 - 1), so for most cases in this sequence, a(n) = n. The first number to achieve a significantly high peak is 138

			a(24)=55 because the aliquot sequence starting at 24 is: 24 - 36 - 55 - 17 - 1 so the maximum peak of this sequence is 55.

Jinyuan Wang, Table of n, a(n) for n = 1..275
W. Creyaufmueller, Aliquot Sequences.
Paul Zimmerman, Aliquot Sequences.

Cf. A003023, A005114, A007906, A037020, A044050, A063769, A098007, A098008, A098009, A098010.

from sympy import divisor_sigma as sigma
def aliquot(n):
    alst = []; seen = set(); i = n
    while i and i not in seen: alst.append(i); seen.add(i); i = sigma(i) - i
    return alst
def aupton(terms): return [max(aliquot(n)) for n in range(1, terms+1)]
print(aupton(66)) # Michael S. Branicky, Jul 11 2021

More terms from Jinyuan Wang, Jul 11 2021

A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15, 13, 14, 17, 18, 21, 19, 20, 22, 23, 24, 36, 55, 25, 26, 27, 28, 29, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 31, 32, 34, 35, 37, 38, 39, 40, 50, 43, 41, 44, 46, 47, 48, 76, 64, 63, 49, 51, 52, 53, 56, 57, 58, 59, 60, 108, 172

Offset: 0

Eric Chen, Sep 13 2021

nonn

This sequence is a permutation of the nonnegative integers iff Catalan's aliquot sequence conjecture (also called Catalan-Dickson conjecture) is true.
a(563) = 276 is the smallest number whose aliquot sequence has not yet been fully determined.
As long as the aliquot sequence of 276 is not known to be finite or eventually periodic, a(563+k) = A008892(k).

			a(0) = 0, a(1) = 1;
since A001065(a(1)) = 0 has already appeared in this sequence, a(2) = 2;
since A001065(a(2)) = 1 has already appeared in this sequence, a(3) = 3;
...
a(11) = 11;
since A001065(a(11)) = 1 has already appeared in this sequence, a(12) = 12;
since A001065(a(12)) = 16 has not yet appeared in this sequence, a(13) = A001065(a(12)) = 16;
since A001065(a(13)) = 15 has not yet appeared in this sequence, a(14) = A001065(a(13)) = 15;
since A001065(a(14)) = 9 has already appeared in this sequence, a(15) = 13;
...

Eric Chen, Table of n, a(n) for n = 0..2703 (all currently known terms, after the b-file of A008892)
Eric Weisstein's World of Mathematics, Aliquot sequence
Eric Weisstein's World of Mathematics, Catalan's Aliquot Sequence Conjecture
Wikipedia, Aliquot sequence

Cf. A032451.
Cf. A001065 (sum of aliquot parts).
Cf. A003023, A044050, A098007, A098008: ("length" of aliquot sequences, four versions).
Cf. A007906.
Cf. A115060 (maximum term of aliquot sequences).
Cf. A115350 (termination of the aliquot sequences).
Cf. A098009, A098010 (records of "length" of aliquot sequences).
Cf. A290141, A290142 (records of maximum term of aliquot sequences).
Cf. A000396, A063990, A122726, A206708, A347770.
Cf. A080907, A063769, A121507, A121508, A131884, A126016.
Aliquot sequences starting at various numbers: A000004 (0), A000007 (1), A033322 (2), A010722 (6), A143090 (12), A143645 (24), A010867 (28), A008885 (30), A143721 (38), A008886 (42), A143722 (48), A143723 (52), A008887 (60), A143733 (62), A143737 (68), A143741 (72), A143754 (75), A143755 (80), A143756 (81), A143757 (82), A143758 (84), A143759 (86), A143767 (87), A143846 (88), A143847 (96), A143919 (100), A008888 (138), A008889 (150), A008890 (168), A008891 (180), A203777 (220), A008892 (276), A014360 (552), A014361 (564), A074907 (570), A014362 (660), A269542 (702), A045477 (840), A014363 (966), A014364 (1074), A014365 (1134), A074906 (1521), A143930 (3630), A072891 (12496), A072890 (14316), A171103 (46758), A072892 (1264460).

A347769_list(N)=print1(0, ", "); if(N>0, print1(1, ", ")); v=[0, 1]; b=1; for(n=2, N, if(setsearch(Set(v), sigma(b)-b), k=1; while(k<=n, if(!setsearch(Set(v), k), b=k; k=n+1, k++)), b=sigma(b)-b); print1(b, ", "); v=concat(v, b))

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A098007 Length of aliquot sequence for n, or -1 if aliquot sequence never cycles.

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A098008 Length of transient part of aliquot sequence for n, or -1 if transient part is infinite.

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A115350 Termination of the aliquot sequence starting at n.

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A007906 Number of steps for aliquot sequence for n to converge to 0, or -1 if it never reaches 0.

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A115060 Maximum peak of aliquot sequence starting at n.

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A347769 a(0) = 0; a(1) = 1; for n > 1, a(n) = A001065(a(n-1)) = sigma(a(n-1)) - a(n-1) (the sum of aliquot parts of a(n-1)) if this is not yet in the sequence; otherwise a(n) is the smallest number missing from the sequence.

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