A008885 -id:A008885 - cp's OEIS frontend

A008890 Aliquot sequence starting at 168.

168, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812

Offset: 0

N. J. A. Sloane

The sum-of-divisor function A000203 and aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 175. - M. F. Hasler, Feb 24 2018

R. K. Guy, Unsolved Problems in Number Theory, B6.

T. D. Noe, Table of n, a(n) for n = 0..175 (a(176) removed by _Georg Fischer_, Apr 28 2019)
Christophe Clavier, Aliquot Sequences
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30), ..., A008892 (starting at 276), A098007 (length of aliquot sequences).

f := proc(n) option remember; if n = 0 then 168; else sigma(f(n-1))-f(n-1); fi; end:

NestList[DivisorSigma[1, #] - # &, 168, 175] (* Alonso del Arte, Feb 24 2018 *)

a(n,a=168)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

Edited by M. F. Hasler, Feb 24 2018

A014363 Aliquot sequence starting at 966.

Original entry on oeis.org

966, 1338, 1350, 2370, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 121770, 241110, 450090, 750870, 1295226, 1572678, 1919538, 2760984, 4964136

Offset: 0

Paul Zimmermann

nonn

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B6, pp. 92-95.

Tyler Busby, Table of n, a(n) for n = 0..1109 (from factordb.com; terms 0..277 from T. D. Noe, terms 278..1035 from Amiram Eldar)
Christophe Clavier, Aliquot Sequences.
factordb.com, Aliquot sequence of 966.
Paul Zimmermann, Latest information.
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30) .. A008892 (starting at 276), A014360 (starting at 552) .. A014365 (starting at 1134), ..., A171103 (starting at 46758), A098007 (length of aliquot sequences).

Cf. A001065.

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0]&, 966, 100] (* Jean-François Alcover, Mar 28 2020 *)

a(n,a=966)={for(i=1,n,a=sigma(a)-a);a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

A014364 Aliquot sequence starting at 1074.

Original entry on oeis.org

1074, 1086, 1098, 1320, 3000, 6360, 13080, 26520, 64200, 136680, 303960, 668040, 1448760, 2897880, 6778920, 14760600, 31761720, 75003840, 189623520, 475142400, 1262108388, 1723154620, 2250655556, 1742856988

Offset: 0

Paul Zimmermann

nonn

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B6, pp. 92-95.

Amiram Eldar, Table of n, a(n) for n = 0..2194 (terms 0..881 from R. J. Mathar)
Christophe Clavier, Aliquot Sequences.
factordb.com, Aliquot sequence of 1074.
Paul Zimmermann, Latest information.
Index entries for sequences related to aliquot parts.

Cf. A098007 (length of aliquot sequences). Some other examples: A008885 (starting at 30) .. A008892 (starting at 276), A014360 (starting at 552) .. A014365 (starting at 1134), ..., A171103 (starting at 46758). See link to index for a more complete list.

Cf. A000203.

FixedPointList[If[# > 0, DivisorSigma[1, #] - #, 0] &, 1074, 100] (* Jean-François Alcover, Mar 28 2020 *)

a(n, a=1074)={for(i=1, n, a=sigma(a)-a); a} \\ M. F. Hasler, Feb 24 2018

a(n+1) = A000203(a(n))-a(n). - R. J. Mathar, Oct 08 2017

A008887 Aliquot sequence starting at 60.

Original entry on oeis.org

60, 108, 172, 136, 134, 70, 74, 40, 50, 43, 1, 0

Offset: 0

N. J. A. Sloane

The sum-of-divisor function A000203 and aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 11. - M. F. Hasler, Feb 24 2018

R. K. Guy, Unsolved Problems in Number Theory, B6.

Christophe CLAVIER, Aliquot Sequences
Index entries for sequences related to aliquot parts.

Cf. A008885 (starting at 30), ..., A008892 (starting at 276), A098007 (length of aliquot sequences).

f := proc(n) option remember; if n = 0 then 60; else sigma(f(n-1))-f(n-1); fi; end:

NestList[If[#==0,0,DivisorSigma[1,#]-#]&,60,80] (* Harvey P. Dale, Nov 29 2013 *)

a(n,a=60)=for(i=1,n,a=sigma(a)-a);a \\ Will raise an error for n > 11, in agreement with the definition. - M. F. Hasler, Feb 24 2018

a(n+1) = A001065(a(n)). - R. J. Mathar, Oct 11 2017

Edited by M. F. Hasler, Feb 24 2018

A032451 Irregular triangle read by rows: there is a row for each value of n for which the aliquot sequence starting at n eventually reaches 1, giving the part of the sequence from n to 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 3, 1, 5, 1, 7, 1, 8, 7, 1, 9, 4, 3, 1, 10, 8, 7, 1, 11, 1, 12, 16, 15, 9, 4, 3, 1, 13, 1, 14, 10, 8, 7, 1, 15, 9, 4, 3, 1, 16, 15, 9, 4, 3, 1, 17, 1, 18, 21, 11, 1, 19, 1, 20, 22, 14, 10, 8, 7, 1, 21, 11, 1, 22, 14, 10, 8, 7, 1, 23, 1, 24, 36, 55, 17, 1, 26, 16

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net)

Related to Catalan's conjecture about iterations of f(n) = sigma(n) - n.

			The aliquot sequences starting with the numbers from 1 to 32 are as follows:
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...]
[7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[8, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[9, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[10, 8, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[12, 16, 15, 9, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143090)
[13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[14, 10, 8, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143721)
[15, 9, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143090)
[16, 15, 9, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143090)
[17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[18, 21, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[20, 22, 14, 10, 8, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143733)
[21, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[22, 14, 10, 8, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143721)
[23, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[24, 36, 55, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143645)
[25, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ...]
[26, 16, 15, 9, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] (A143759)
[27, 13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, ...]
[29, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0, 0, 0, 0, 0, 0, ...] (A008885)
[31, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
[32, 31, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...]
Rows 6, 25 and 28 are omitted from the entry since they never reach 1.

L. Alaoglu and P. Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), 881-882.

with(numtheory);
f:=proc(n) local i,t1; t1:=[n];
for i from 1 to 20 do t1:=[op(t1), sigma(t1[i])-t1[i]]; od:
t1; end;
for n from 2 through 32 do lprint(f(n)); od:

Edited by N. J. A. Sloane, Nov 30 2008

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))

A362440 Aliqout sequence starting at 841.

Original entry on oeis.org

841, 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0

Offset: 1

N. J. A. Sloane, Apr 22 2023

This is 841 followed by A008885. Could be further extended backwards, see A070015.

GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 92.

Cf. A001065, A008885, A070015.

NestWhileList[DivisorSigma[1,#]-#&,841,#>0&] (* Paolo Xausa, Oct 16 2023 *)

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A008890 Aliquot sequence starting at 168.

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A014363 Aliquot sequence starting at 966.

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A014364 Aliquot sequence starting at 1074.

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A008887 Aliquot sequence starting at 60.

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A032451 Irregular triangle read by rows: there is a row for each value of n for which the aliquot sequence starting at n eventually reaches 1, giving the part of the sequence from n to 1.

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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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A362440 Aliqout sequence starting at 841.

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Mathematica