A098008 -id:A098008 - cp's OEIS frontend

A234899 Record holders for lengths of ever-decreasing aliquot sequences.

1, 2, 4, 9, 14, 16, 26, 46, 52, 166, 212, 1113, 2343, 4437, 5145, 8535, 10665, 18711, 33682, 64935, 114808, 187232, 228316, 304412, 464132, 556636, 623288, 1230284, 1319956, 1508504, 2897884, 3835556, 7487494, 9446906, 16871648, 22328212, 29668150, 29725184

Offset: 1

Michel Marcus, Jan 01 2014

nonn

If one looks at the lengths of uninterrupted decreasing aliquot sequences, the converse of A081705, one gets a sequence similar to A098008, except for perfect or abundant numbers, but also for numbers that encounter a perfect or abundant numbers in this process.
The current sequence lists the deficient numbers yielding uninterrupted decreasing aliquot sequences that are longer than any previous ones (compare with A081699).
Note that, so far, the lengths of the corresponding sequences are contiguous. Does it remain so for next terms?

			The aliquot sequence starting at 2 decreases as follows 2->1->0 and is longer than the sequence starting at 1. Hence 2 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..59

Cf. A081699, A081705, A098008.

nbdecr(n) = {nb = 0; while (n && ((newn = sigma(n)-n)) < n, n = newn ; nb++); nb;}
lista(nn) = {recab = 0; for (ni = 1, nn, ab = nbdecr(ni); if (ab > recab, recab = ab; print1(ni, ", ")););}

A290141 Numbers n that have a record maximum (> n) in their aliquot sequence.

Original entry on oeis.org

12, 18, 20, 24, 30, 102, 120, 138

Offset: 1

Amiram Eldar, Jul 21 2017

Maximum term in the aliquot sequence of n is considered only if it is larger than n.

The record values are in A290142.

			The aliquot sequence of 30 is: 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1. The maximum is 259 which is larger than 30, and larger than the maxima of all the aliquot sequences of the numbers below 30.

Cf. A098008, A098009, A098010, A290142.

g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; seq = {}; a = -1; seq = {}; Do[b = Max[Drop[f[n], 1]];  If[b > a, a = b; AppendTo[seq, n]], {n, 2, 275}]; seq (* after Robert G. Wilson v at A098009 *)

A290142 Records of the maxima of the aliquot sequences of the numbers in A290141.

Original entry on oeis.org

16, 21, 22, 55, 259, 759, 32571, 179931895322

Offset: 1

Amiram Eldar, Jul 21 2017

a(8) was calculated by D. H. Lehmer.

			The aliquot sequence of 30 is: 30, 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1. The maximum is 259 which is larger than 30, and larger than the maxima of all the aliquot sequences of the numbers below 30.

Cf. A008888, A098008, A098009, A098010, A290141.

g[n_] := If[n > 0, DivisorSigma[1, n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; rec = {}; a = -1; seq = {}; Do[b = Max[Drop[f[n], 1]];
If[b > a, a = b; AppendTo[rec, b]], {n, 2, 275}] ; rec (* after Robert G. Wilson v at A098009 *)

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

A290143 Numbers n such that transient part of the unitary aliquot sequence for n sets a new record.

Original entry on oeis.org

1, 2, 10, 14, 22, 38, 70, 134, 138, 170, 190, 210, 318, 426, 1398, 4170, 6870, 8454, 19866, 22470, 36282, 38370, 70770, 84774, 98790, 132990, 474642, 705990, 961650

Offset: 1

Amiram Eldar, Jul 21 2017

The unitary version of A098009.

The record values are in A290144.

			The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.

Richard K. Guy, "Unitary aliquot sequences", Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. B8, pp. 97-99.
Richard K. Guy and Marvin C. Wunderlich, Computing Unitary Aliquot Sequences: A Preliminary Report, University of Calgary, Department of Mathematics and Statistics, 1979.
H. J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
H. J. J. te Riele, Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Eric Weisstein's World of Mathematics, Unitary Aliquot Sequence

Cf. A098008, A098009, A098010, A127652, A127653, A127654, A127655, A290144.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; seq = {}; Do[b = Length[f[n]] - 1; If[b > a, a = b; AppendTo[seq, n]], {n, 10^6}] ; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)

A290144 Record lengths of transient part of the unitary aliquot sequences of the numbers in A290143.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 43, 45, 67, 78, 205, 207, 1109, 1116, 1117, 1155, 1162, 1163, 1171, 1711, 1712, 1828, 1829

Offset: 1

Amiram Eldar, Jul 21 2017

			The unitary aliquot sequence of 134 is: 134, 70, 74, 40, 14, 10, 8, 1. Its length is 8 and it is longer than the unitary aliquot sequences of all the numbers below 134.

Cf. A098008, A098009, A098010, A127652, A127653, A127654, A127655, A290143.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = 0; seq = {}; Do[b = Length[f[n]] - 2; If[b > a, a = b; AppendTo[seq, b]], {n, 10^6}]; seq (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)

A290145 Numbers n that have a record maximum in their unitary aliquot sequence.

Original entry on oeis.org

30, 66, 102, 138, 174, 210, 318, 1110, 1398, 6870, 19866, 89610, 291450, 705990

Offset: 1

Amiram Eldar, Jul 21 2017

Maximum term in the aliquot sequence of n is considered only if it is larger than n.
The record values are in A290146.
te Riele found the unitary aliquot sequences of all numbers < 10^5, except for 89610. He terminated the calculation of the unitary aliquot sequence of 89610 at the 541st iteration, at 114601234388928504726, while the maximum, 645856907610421353834, is reached at the 569th iteration.

			The unitary aliquot sequence of 174 is: 174, 186, 198, 162, 84, 76, 24, 12, 8, 1. Its maximum is 198 which larger than the maxima of all the aliquot sequences of the numbers below 174.

H. J. J. te Riele, Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.

Cf. A098008, A098009, A098010, A290146.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; rec = {}; Do[b = Length[f[n]] - 2;
If[b > a, a = b; AppendTo[rec, n]], {n, 10^6}] ; rec (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)

A290146 Records of the maxima of the unitary aliquot sequences of the numbers in A290145.

Original entry on oeis.org

54, 90, 126, 162, 198, 378, 4950, 12978, 82278, 94218606, 8855754260391450, 645856907610421353834, 1350253136232108126126, 27709820863862780667438

Offset: 1

Amiram Eldar, Jul 21 2017

			The unitary aliquot sequence of 174 is: 174, 186, 198, 162, 84, 76, 24, 12, 8, 1. Its maximum is 198 which larger than the maxima of all the aliquot sequences of the numbers below 174.

Cf. A098008, A098009, A098010, A290145.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])];
g[n_] := If[n > 0, usigma[n] - n, 0]; f[n_] := NestWhileList[g, n, UnsameQ, All]; a = -1; rec = {}; Do[b = Length[f[n]] - 2; If[b > a, a = b; AppendTo[rec, b ]], {n, 10^6}] ; rec (* after Giovanni Resta at A034448 & Robert G. Wilson v at A098009 *)

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))

cp's OEIS Frontend

A234899 Record holders for lengths of ever-decreasing aliquot sequences.

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PARI

A290141 Numbers n that have a record maximum (> n) in their aliquot sequence.

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A290142 Records of the maxima of the aliquot sequences of the numbers in A290141.

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Mathematica

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

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PARI

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

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Maple

A290143 Numbers n such that transient part of the unitary aliquot sequence for n sets a new record.

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A290144 Record lengths of transient part of the unitary aliquot sequences of the numbers in A290143.

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A290145 Numbers n that have a record maximum in their unitary aliquot sequence.

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A290146 Records of the maxima of the unitary aliquot sequences of the numbers in A290145.

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