A098007 -id:A098007 - cp's OEIS frontend

A216072 Aliquot open end sequences which belong to distinct families.

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.

A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

Original entry on oeis.org

1264460, 1305184, 1547860, 1727636, 4938136, 5423384, 5504056, 5753864, 18656380, 20522060, 24289964, 28158165, 28630036, 29902635, 29971755, 30853845, 81128632, 91314968, 91401368, 96389032, 209524210, 230143790, 231439570, 246667790, 498215416, 506040584, 510137384, 583014136

Offset: 1

Zoltan Galantai, Sep 11 2017

This list is not known to be complete (564 might be a member). See A122726. - N. J. A. Sloane, Sep 17 2021

Up to the known 1593 sociable number cycles, 96.1% of the sociable number cycles satisfy this condition (up to the first 10 sociable number cycles: 40%; up to the first 100 sociable number cycles: 77%; up to the first 500 sociable number cycles: 92%, and up to the first 1000 sociable number cycles: 94.9%). So the conjecture here is that as the number of sociable number cycles increases, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%. Notice that the sums of amicable pairs are similarly often divisible by 10, but are not included here (see A291422).

			The sum of 1264460, 1547860, 1727636 and 1305184 is divisible by ten, thus this sociable number cycle belongs to the sequence. On the other hand, the 12496, 14288, 15472, 14536, 14264 sociable number cycle does not qualify since its sum is 71506.

R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, 1994, pp. 62 - 63.
Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chappman and HALL/CRC, 2003, pp. 2747 - 2748.
Song Y. Yan, Perfect, Amicable and Sociable Numbers. A Computation Approach, World Scientific 1996, pp. 34 - 38.

Zoltan Galantai, Table of n, a(n) for n = 1..36
Zoltan Galantai, List of known sociable number cycles where the sums of the cycles is divisible by 10.
Shyam Sunder Gupta, Perfect, Multiply Perfect, and Sociable Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 6, 185-207.
David Moews, A list of currently known aliquot cycles of length greater than 2 [This list is not known to be complete.]
Eric Weisstein's World of Mathematics, Sociable Numbers

Cf. A122726, A003416, A090615, A119478, A098007, A126016, A063990, A291422.

Changed definition and added comment to point out that this sequence is only conjectural. - N. J. A. Sloane, Sep 17 2021

A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Apr 12 2024

Analogous to A098007 with A033880(n) = sigma(n) - 2*n instead of A001065(n) = sigma(n) - n.

			a(n) = 0 if the iterations that start at n are entering a cycle. Examples of cycles are:
  1) Cycles of length 1: the triperfect numbers (A005820), 120, 672, 523776, ..., which are the fixed points of A033880. The triperfect numbers can be reached from other values of n, e.g., 276, 448, 486, 510, 702, ... .
  2) Cycles of length 2: the only known cycle is (45840, 51168) (see A069085). It can be reached from other values of n, e.g., 32130, 39420, 45480, 66300, ... .
  3) Cycles of length 3: the least cycle is (243732672, 271303776, 256786848). It is first reached from n = 107689320.
  4) Cycles of length 4: the least cycle is (65071776, 82842816, 89761152, 77260656). It can be reached from other values of n, e.g., 33623940, 41132280, 42825888, ... . The next cycle of length 4 is (985948800, 1381340160, 2183133696, 1489384608).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A001065, A033880, A098007.

Cf. A005101, A005820, A069085, A113285, A263837, A371920.

ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; a[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; Array[a, 120]

ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k));}
a(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<M. F. Hasler at A098007

a(n) = 1 if and only if n is nonabundant (A263837).
If a(n) > 0 then:
  a(n) > 1 if n is abundant (A005101).
  a(n) > 2 if n is in A371920.

A234842 Primes that are reached by an ever increasing aliquot sequence.

Original entry on oeis.org

463, 523, 983, 1153, 2851, 2969, 4339, 4507, 6121, 8263, 8893, 10093, 12451, 17911, 18427, 18913, 22807, 22811, 25033, 27961, 33223, 36781, 41849, 42643, 48571, 60091, 64237, 71503, 73303, 74131, 90217, 90481, 103813, 108263, 123601, 124447, 125863, 140443

Offset: 1

Michel Marcus, Dec 31 2013

nonn

Note that the starting point of these aliquot sequences are not in increasing order, since for instance we have: 392->463->1 and 324->523->1, that is, with 392>324 while 463<523.
One can observe that the "ever increasing aliquot" part in the definition is not really necessary. A prime is in the sequence if there is an abundant number whose sum of proper divisors results into this prime. So sequence could also be defined as: Primes resulting from summing up the proper divisors of an abundant number. - Michel Marcus, Jan 05 2014
If we try to build the revert sequence listing the starting points of the aliquot sequences, we would get the following terms in increasing order 324, 392, 784, 800, 2304, 2450, 2704, 3600, 3872. But then for n=5352, we'd hit a sequence that begins 5352->8088->12192->20064 and keeps rising to a point where the factors of the last known term are not known. Then later, there are several other such aliquot sequences like 9336->14064->22392 or 10344->15576->27624 that have the same behavior. Thus the only sure terms of the revert sequence would be the terms listed earlier. - Michel Marcus, Jan 11 2014

			The aliquot sequence that begins with 10712 is always increasing before reaching prime 12451: 10712->11128->11552->12451->1, hence 12451 is in the sequence.
20422951 also belongs here with the aliquot sequence that starts at 14952, so a 13-tuple abundant (see factordb link).
People at the Aliquot Sequences project have found longer sequences that reach higher primes.

Michel Marcus, Table of n, a(n) for n = 1..100
Factordb, Sequence starting with 14952
Michel Marcus, Aliquot sequences leading to these primes
Help needed for a sequence Thread on Aliquot Sequences Forum

Cf. A001065, A005101, A080907, A081705, A098007, A098008, A115350.

prev(n) = {for (i=1, n, if ((sigma(i) - i) == n, return (i));); return (0);}
lista(nn) = {forprime(p=2, nn, if (prev(p), print1(p, ", ");););} \\ simplified by Michel Marcus, Jan 11 2014

A254618 a(n) = k-tuple deficiency of n-th deficient number.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 4, 4, 2, 2, 5, 5, 6, 2, 2, 3, 6, 2, 1, 7, 3, 2, 2, 3, 6, 1, 3, 2, 7, 3, 2, 2, 1, 7, 8, 2, 4, 3, 4, 9, 2, 3, 3, 4, 2, 2, 2, 3, 4, 3, 2, 5, 4, 2, 2, 1, 5, 5, 3, 2, 1, 2, 2, 3, 9, 7, 2, 4, 6, 4, 4, 2, 2, 3, 4, 2, 2, 8, 1, 2, 2, 2, 3, 2, 3, 5

Offset: 1

Paolo P. Lava, Feb 03 2015

nonn

For any deficient number x iterate the process f(x)=sigma(x)-x. Sequence lists how many times f(x) keeps deficient until it reaches zero.
Non-deficient numbers are excluded from this sequence.
k-tuple deficiency records is A000027.
k-tuple deficiency record-holders is A234899.

			a(20) = 1 because the 20th deficient number is 25 and:
1) f(25) = sigma(25) - 25 = 6 < 25.
We must stop here because 6 is abundant.
a(21) = 7 because the 21st deficient number is 26 and:
1) f(26) = sigma(26) - 26 = 16 < 26;
2) f(16) = sigma(16) - 16 = 15 < 16;
3) f(15) = sigma(15) - 15 = 9 < 15;
4) f(9) = sigma(9) - 9 = 4 < 9;
5) f(4) = sigma(4) - 4 = 3 < 4;
6) f(3) = sigma(3) - 3 = 2 < 1;
7) f(1) = sigma(1) - 1 = 0 < 1.
We must stop here because sigma(0) is not defined.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000027, A005100, A081705, A098007, A098008, A234899.

with(numtheory): P:=proc(q) local a,b,n,t;
for n from 1 to q do t:=0; b:=sigma(n)-n; a:=n;
if b

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))

A371922 Indices of records in A371921: numbers k such that A371921(k) > A371921(m) for all m < k.

Original entry on oeis.org

1, 12, 24, 90, 204, 330, 540, 1080, 1140, 2184, 3480, 6324, 15630, 23496, 38340, 48510, 56760, 99636, 234960, 270180, 300150, 528180, 703080, 973644, 1907178, 5992380, 7980930, 12032640, 20687436, 23847642, 27465840, 28653720, 34964340

Offset: 1

Amiram Eldar, Apr 12 2024

The corresponding record values are 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 13, ... (see the link for more values).

			The iterations of A033880 over the first 8 terms:
 n | a(n) | Iterations
 --+------+----------------------------------------------------------------
 1 |    1 | 1 -> -1
 2 |   12 | 12 -> 4 -> -1
 3 |   24 | 24 -> 12 -> 4 -> -1
 4 |   90 | 90 -> 54 -> 12 -> 4 -> -1
 5 |  204 | 204 -> 96 -> 60 -> 48 -> 28 -> 0
 6 |  330 | 330 -> 204 -> 96 -> 60 -> 48 -> 28 -> 0
 7 |  540 | 540 -> 600 -> 660 -> 696 -> 408 -> 264 -> 192 -> 124 -> -24
 8 | 1080 | 1080 -> 1440 -> 2034 -> 378 -> 204 -> 96 -> 60 -> 48 -> 28 -> 0

Amiram Eldar, Table of n, a(n), A371921(a(n)) for n = 1..33.

Cf. A033880, A098007, A371921.

ab[n_] := Module[{k}, If[n < 1, 0, k = DivisorSigma[1, n] - 2*n; If[k < 1, 0, k]]]; f[n_] := Module[{s = NestWhileList[ab, n, UnsameQ, All]}, If[s[[-1]] == 0, Length[s] - 2, 0]]; seq[max_] := Module[{fm = 0, f1, s = {}}, Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, max}]; s]; seq[10^5]

ab(n) = {my(k); if(n < 1, 0, k = sigma(n) - 2*n; if(k < 1, 0, k));}
f(n) = {my(t = 0); until(bittest(t, n = ab(n)), t += 1<M. F. Hasler at A098007
lista(kmax) = {my(fm = 0, f1); for(k = 1, kmax, f1 = f(k); if(f1 > fm, fm = f1; print1(k, ", "))); }

A347037 The length of the sequence before a repeated number appears, or -1 if a repeat never occurs, starting at k = n for the iterative cycle k -> sigma(k) - k if k is even, k -> sigma(k) if k is odd, where sigma(k) = sum of divisors of k.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 6, 3, 245, 244, 5, 4, 242, 243, 5, 4, 7, 6, 3, 5, 242, 241, 3, 244, 5, 1, 22, 21, 2, 8, 7, 8, 240, 7, 6, 6, 4, 21, 20, 4, 18, 245, 8, 7, 8, 4, 239, 246, 20, 19, 239, 5, 7, 3, 12, 11, 9, 8, 5, 5, 468, 18, 5, 4, 471, 6, 239, 238, 6, 5, 13, 6, 471, 17, 7, 6, 14, 5, 468

Offset: 1

Scott R. Shannon, Aug 12 2021

nonn

Various starting values, which are even or reach an even value, follow the associated trajectory of the standard aliquot sequence A098007 and thus grow to values whose ultimate fate is currently unknown. The first such term is 165 as its cycle begins 165 -> 288 -> 531 -> 780, from which point it follows the aliquot sequence starting at 780 that after 3485 further cycles reaches a term with 198 digits whose factorization is not known.
This sequence also has the same loops seen in the aliquot sequence. All even perfect numbers repeat after one value, and all even-numbered amicable pairs and even-numbered sociable number loops are also present. The amicable pair 2620-2934 is first reached from n = 477 after 329 cycles while the pair 1184-1210 is first reached from n = 847 after 4 cycles. The 28-member sociable number loop containing the number 376736, see A072890, is first reached from n = 267 after 52 cycles.
Loops present in this sequence that are not in the aliquot sequence are two-member loops formed by a Mersenne prime and the even number, a power of 2, one greater than it. For starting values n <= 1000 two other loops are present which end numerous cycles. One is a five-member loop comprising the numbers 56 -> 64 -> 63 -> 104 -> 106 (-> 56), first seen for starting value n = 11. The other is a four-member loop comprising the numbers 40 -> 50 -> 43 -> 44 (-> 40), first seen for starting value n = 27. It is likely other such loops exist for starting values >> 1000.
As sigma(prime) = prime + 1, such prime starting values will have the same cycle, with one extra term, as the later even term, assuming the prime is not a Mersenne prime.
For the first 1000 terms the longest currently known cycle if one of 1538 steps, starting at n = 693 and ending with 104. The largest value reached in this series is a 58-digit number 12316...65104. The majority of this series follows the 1074-step aliquot sequence starting at 1248.

			a(3) = 2 as 3 -> 4 -> 3. Similarly for all other Mersenne primes.
a(5) = 2 as 5 -> 6 -> 6. Similarly for all other primes one less than a perfect number.
a(6) = 1 as 6 -> 6. Similarly for all other even perfect numbers.
a(11) = 245 as 11 -> 12 -> 16 -> 15 -> 24 -> (233 more terms) -> 230 -> 202 -> 104 -> 106 -> 56 -> 64 -> 63 -> 104, ending with the five-member loop.
a(19) = 7 as 19 -> 20 -> 22 -> 14 -> 10 -> 8 -> 7 -> 8.
a(27) = 5 as 27 -> 40 -> 50 -> 43 -> 44 -> 40, ending with the four-member loop.
a(847) = 5 as 847 -> 1064 -> 1336 -> 1184 -> 1210 -> 1184, ending with the second smallest amicable pair.

Wikipedia, Aliquot sequence
P. Zimmermann, Aliquot Sequences

Cf. A098007, A001065, A259180, A003416, A072890, A000668.

Table[t=k=0;lst={n};k=If[OddQ@n,DivisorSigma[1,n],DivisorSigma[1,n]-n];While[FreeQ[lst,k],AppendTo[lst,k];n=k;t++;k=If[OddQ@n,DivisorSigma[1,n],DivisorSigma[1,n]-n]];t+1,{n,100}] (* Giorgos Kalogeropoulos, Aug 14 2021 *)

cp's OEIS Frontend

A216072 Aliquot open end sequences which belong to distinct families.

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A292217 Conjectured list of numbers in increasing order that belong to sociable cycles of length greater than 2 in which the sum of the cycle is divisible by 10.

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A371921 The number of iterations of the map x -> A033880(x) starting at n until the a nonpositive number is reached, or 0 if this does not happen.

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A234842 Primes that are reached by an ever increasing aliquot sequence.

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A254618 a(n) = k-tuple deficiency of n-th deficient number.

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Maple

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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A371922 Indices of records in A371921: numbers k such that A371921(k) > A371921(m) for all m < k.

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A347037 The length of the sequence before a repeated number appears, or -1 if a repeat never occurs, starting at k = n for the iterative cycle k -> sigma(k) - k if k is even, k -> sigma(k) if k is odd, where sigma(k) = sum of divisors of k.

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