A115350 -id:A115350 - cp's OEIS frontend

A008885 Aliquot sequence starting at 30.

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

Offset: 0

The sum-of-divisor function A000203 and thus aliquot parts A001065 are defined only for positive integers, so the trajectory ends when 0 is reached, here at index 15. - M. F. Hasler, Feb 24 2018

			a(0) = 30.
30 has eight divisors, 1, 2, 3, 5, 6, 10, 15, 30, which add up to 72, and 72 - 30 = 42, so a(1) = 42.

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

Christophe CLAVIER, Aliquot Sequences
R. J. Mathar, Illustration of Aliquot Sequence Mergers (2014) Fig 1 (prime family 3)
Index entries for sequences related to aliquot parts.

Cf. A008886, A008887, A008888, A008889, A008890, A008891, A008892.

f := proc(n) option remember; if n = 0 then 30; else sigma(f(n-1))-f(n-1); fi; end:

NestList[If[# > 0, DivisorSigma[1, #] - #, 0] &, 30, 80] (* Harvey P. Dale, Jun 12 2012 *)

a(n, a=30)=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

A063769 Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.

Original entry on oeis.org

25, 95, 119, 143, 417, 445, 565, 608, 650, 652, 675, 685, 783, 790, 909, 913

Offset: 1

Tanya Khovanova and Alexey Radul, Aug 14 2001

There are many numbers whose aliquot sequences have not yet been completely computed, so this sequence is not fully known. In particular, 276 may, perhaps, be an element of this sequence, although this is very unlikely.

Numbers less than 1000 whose aliquot sequence is not known that could possibly be in this sequence are: 276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996. - Robert Price, Jun 03 2013

			The divisors of 95 less than itself are 1, 5 and 19. They sum to 25. The divisors of 25 less than itself are 1 and 5. They sum to 6, which is perfect.

No number terminates at 28, the second perfect number.

Eric Weisstein's World of Mathematics, Aspiring Number

Cf. A080907, A115350.

perfectQ[n_] := DivisorSigma[1, n] == 2*n; maxAliquot = 10^45; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = False; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] // perfectQ; Reap[For[k = 1, k < 1000, k++, If[! perfectQ[k] && selQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Nov 15 2013 *)

a(13)-a(16) from Robert Price, Jun 03 2013

A080907 Numbers whose aliquot sequence terminates in a 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Gabriel Cunningham (gcasey(AT)mit.edu), Mar 31 2003

All primes are in this set because s(p) = 1 for p prime. Perfect numbers are clearly not in this set. Neither are aspiring numbers (A063769), or numbers whose aliquot sequence is a cycle (such as 220 and 284).

There are some numbers whose aliquot sequences haven't been fully determined (such as 276).

			4 is in this set because its aliquot chain is 4->3->1. 6 is not in this set because it is perfect. 25 is not in this set because its aliquot chain is 25->6.

Jean-François Alcover, Table of n, a(n) for n = 1..1000
M. Benito, W. Creyaufmüller, J. L. Varona, and P. Zimmermann, Aliquot Sequence 3630 Ends After Reaching 100 Digits, Experimental Mathematics (2002), Volume 11, issue 2.
Markus Tervooren, Aliquot sequence of 840
Eric Weisstein's World of Mathematics, Aliquot Sequence

Cf. A063769, A115350, A131884.

Complement of A126016.

maxAliquot = 10^50; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print["A131884: ", n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] & , All] == 1; Select[Range[1, 1100], selQ] (* Jean-François Alcover, Nov 14 2013, updated Sep 10 2015 *)

n is a member if n = 1 or s(n) is a member, where s(n) is the sum of the proper factors of n.

Edited by N. J. A. Sloane, Aug 14 2006
More terms from Franklin T. Adams-Watters, Dec 14 2006
The fact that 840 was missing from the sequence b-file was pointed out by Philip Turecek, Sep 10 2015

A131884 Numbers conjectured to have an infinite, aperiodic, aliquot sequence.

Original entry on oeis.org

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836

Offset: 1

J. Lowell, Oct 24 2007

From Martin Renner, Oct 28 2011: (Start)
There are 12 numbers up to 1000 with the five yet unknown trajectories
(1) 276 ->
    306 -> 396 -> 696 -> ...
(2) 552 -> 888 -> ...
(3) 564 -> 780 -> ...
(4) 660 ->
    828 ->
    996 -> 1356 -> ...
(5) 966 -> 1338 -> ...
The least starting numbers 276, 552, 564, 660 and 966 for the trajectories are called Lehmer five.
There are currently 81 open end trajectories up to 10000. (End)
Sequence A216072 lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. But this sequence lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. - V. Raman, Dec 08 2012

Christophe Clavier: Aliquot sequences (with leading term < 10,000).
Wolfgang Creyaufmüller: Primzahlfamilien - aliquot sequences.
Paul Zimmermann: Aliquot sequences.

Cf. A098007, A216072, A008892, A115350.

(* This script is not suitable for a large number of terms *) maxAliquot = 10^50; A131884 = {}; s[1] = 1; s[n_] := DivisorSigma[1, n] - n; selQ[n_ /; n <= 5] = True; selQ[n_] := NestWhile[s, n, If[{##}[[-1]] > maxAliquot, Print[n]; AppendTo[A131884, n]; False, Length[{##}] < 4 || {##}[[-4 ;; -3]] != {##}[[-2 ;; -1]]] &, All] == 1; selQ /@ Range[1000]; A131884 (* Jean-François Alcover, Sep 10 2015 *)

More terms and links from Martin Renner, Oct 28 2011

A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

Original entry on oeis.org

2, 3, 6, 21, 37, 31, 49, 79, 73, 91, 115, 127, 151, 121, 181, 169, 217, 265, 253, 271, 211, 301, 433, 379, 331, 361, 457, 391, 451, 655, 463, 541, 421, 775, 511, 769, 673, 715, 865, 691, 1015, 631, 1069, 1075, 721, 931, 781, 1123, 871, 925, 901, 1177, 991, 1297

Offset: 0

Klaus Brockhaus, Nov 27 2006

nonn

Minimal values for nodes of exact degree in aliquot sequences. Find each node's degree (number of predecessors) in aliquot sequences and choose the smallest value as the sequence member. - Ophir Spector, ospectoro (AT) yahoo.com Nov 25 2007

			a(4) = 37 since there are exactly four numbers (155, 203, 299, 323) whose sum of proper divisors is 37. For k < 37 there are either fewer or more numbers (32, 125, 161, 209, 221 for k = 31) whose sum of proper divisors is k.

Daniel Mondot, Table of n, a(n) for n = 0..5646 (First 157 terms from Ophir Spector. First 1000 terms from Ophir Spector and Donovan Johnson.)
W. Creyaufmueller, Aliquot sequences
Daniel Mondot, A-file, extended version of b-file but with some gaps towards the end.
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]
Eric Weisstein's World of Mathematics, Aliquot sequence

Cf. A001065, A048138, A070015, A123930, A080907, A115350, A121507, A037020, A126016, A057709, A057710, A063990.

{m=54;z=1500;y=600000;v=vector(z);for(n=2,y,s=sigma(n)-n; if(s

A127163 Integers whose aliquot sequences terminate by encountering the prime 3. Also known as the prime family 3.

Original entry on oeis.org

3, 4, 9, 12, 15, 16, 26, 30, 33, 42, 45, 46, 52, 54, 66, 72, 78, 86, 87, 90, 102, 105, 114, 121, 123, 126, 135, 144, 165, 166, 174, 186, 198, 207, 212, 243, 246, 247, 249, 258, 259, 270

Offset: 1

Ant King, Jan 07 2007

This sequence is complete only as far as the last term given, for the eventual fate of the aliquot sequence generated by 276 is not (yet) known

			a(5)=15 because the fifth integer whose aliquot sequence terminates by encountering the prime 3 as a member of its trajectory is 15. The complete aliquot sequence generated by iterating the proper divisors of 15 is 15->9->4->3->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127161, A127162, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 3] &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) terminates by encountering the prime 3 as a member of its trajectory, i is included in this sequence

A127164 Integers whose aliquot sequences terminate by encountering the prime 7. Also known as the prime family 7.

Original entry on oeis.org

7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, 152, 169, 188, 213, 215

Offset: 1

Ant King, Jan 07 2007

This sequence is complete only as far as the last term given, for the eventual fate of the aliquot sequence generated by 276 is not (yet) known.

			a(5)=20 because the fifth integer whose aliquot sequence terminates by encountering the prime 7 as a member of its trajectory is 20. The complete aliquot sequence generated by iterating the proper divisors of 15 is 20->22->14->10->8->7->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127161, A127162, A127163, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 7] &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) terminates by encountering the prime 7 as a member of its trajectory, i is included in this sequence.

A127161 Integers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Ant King, Jan 06 2007

nonn

This sequence is the same as A080907 from A080907's second term onwards.

			a(10)=12 because the tenth integer whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127162, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], Last[Trajectory[ # ]] == 0 &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly iterating s contains a prime as a member of its trajectory, i is included in this sequence

A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.

Original entry on oeis.org

4, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 96, 98, 99

Offset: 1

Ant King, Jan 06 2007

nonn

			a(5)=12 because the fifth composite number whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0

Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.

Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
Wolfgang Creyaufmueller, Aliquot sequences.

Cf. A080907, A127161, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], ! PrimeQ[ # ] && Last[Trajectory[ # ]] == 0 &]

Define s(i)=sigma(i)-i=A000203(i)-i. Then if i is composite and the aliquot sequence obtained by repeatedly applying the mapping i->s(i) contains a prime as a member of its trajectory, i is included in this sequence.

A216072 Aliquot open end sequences which belong to distinct families.

Original entry on oeis.org

276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842

Offset: 1

V. Raman, Sep 01 2012

nonn

These aliquot sequences are believed to grow forever without terminating in a prime or entering a cycle.

Sequence A131884 lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. But this sequence lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. - V. Raman, Dec 08 2012

V. Raman, Table of n, a(n) for n = 1..81
Christophe Clavier, Aliquot Sequences with leading terms < 10000
Wolfgang Creyaufmüller, Aliquot Sequences
Paul Zimmermann, Aliquot Sequences.

Cf. A008892, A098007, A131884, A115350.

cp's OEIS Frontend

A008885 Aliquot sequence starting at 30.

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A063769 Aspiring numbers: numbers whose aliquot sequence terminates in a perfect number.

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A080907 Numbers whose aliquot sequence terminates in a 1.

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A131884 Numbers conjectured to have an infinite, aperiodic, aliquot sequence.

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A125601 a(n) is the smallest k > 0 such that there are exactly n numbers whose sum of proper divisors is k.

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A127163 Integers whose aliquot sequences terminate by encountering the prime 3. Also known as the prime family 3.

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A127164 Integers whose aliquot sequences terminate by encountering the prime 7. Also known as the prime family 7.

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