examples of aliquot sequences

examples of aliquot sequences

The shortest

aliquot sequence

with distinct

elements

is that for 1, simply: 1, 0.

A

prime number

p

>

1

has an aliquot sequences that is almost as short, namely:

p

, 1, 0. For example, 47, 1, 0.

Technically, all aliquot sequences are

infinite

, but some cease to be interesting sooner than others. Throughout this article, length will refer to the length of the aliquot sequence from its first element up to the first

instance

of the

fixed point

.

Some aliquot sequences zag all over the place before finally settling down on 0. The aliquot sequence for 60, for example: 60, 108, 172, 136, 134, 70, 74, 40, 50, 43, 1, 0. Or for a more dramatic example: 138, 150, 222, 234, 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, 2109796, 1889486, 953914, 668966, 353578, 176792, 254128, 308832, 502104, 753216, 1240176, 2422288, 2697920, 3727264, 3655076, 2760844, 2100740, 2310856, 2455544, 3212776, 3751064, 3282196, 2723020, 3035684, 2299240, 2988440, 5297320, 8325080, 11222920, 15359480, 19199440, 28875608, 25266172, 19406148, 26552604, 40541052, 54202884, 72270540, 147793668, 228408732, 348957876, 508132204, 404465636, 303708376, 290504024, 312058216, 294959384, 290622016, 286081174, 151737434, 75868720, 108199856, 101437396, 76247552, 76099654, 42387146, 21679318, 12752594, 7278382, 3660794, 1855066, 927536, 932464, 1013592, 1546008, 2425752, 5084088, 8436192, 13709064, 20563656, 33082104, 57142536, 99483384, 245978376, 487384824, 745600776, 1118401224, 1677601896, 2538372504, 4119772776, 8030724504, 14097017496, 21148436904, 40381357656, 60572036544, 100039354704, 179931895322, 94685963278, 51399021218, 28358080762, 18046051430, 17396081338, 8698040672, 8426226964, 6319670230, 5422685354, 3217383766, 1739126474, 996366646, 636221402, 318217798, 195756362, 101900794, 54202694, 49799866, 24930374, 17971642, 11130830, 8904682, 4913018, 3126502, 1574810, 1473382, 736694, 541162, 312470, 249994, 127286, 69898, 34952, 34708, 26038, 13994, 7000, 11720, 14740, 19532, 16588,
18692, 14026, 7016, 6154, 3674, 2374, 1190, 1402, 704, 820, 944, 916, 694, 350, 394, 200, 265, 59, 1, 0.

Despite its length, 138’s aliquot sequence does

eventually

settle down. One doesn’t have to look too far up to find a much, much longer aliquot sequence: 276, 396, 696, 1104, 1872, 3770, 3790, 3050, 2716, 2772, 5964, 10164, 19628, 19684, 22876, 26404, 30044, 33796, 38780, 54628, 54684, 111300, 263676, 465668, 465724, 465780, 1026060, 2325540, 5335260, 11738916, 23117724, 45956820, 121129260, 266485716, 558454764, 1092873236, 1470806764, 1471882804, 1642613196, 2737688884, 2740114636, 2791337780, 4946860492, 4946860548, 9344070652, 9344070708, 15573451404, 27078171764, 27284104204, 27410152084, 27410152140, 76787720100, 220578719452, 254903331620, 361672366300, 603062136740, 921203207260, 1381419996068, 1395444575644, 1395478688996, 1395546402460, 2069258468900, 3065057872156, 3277068463844,
3429776547484, 3597527970596, 4028517592540, 5641400009252, 5641400009308, 5641400009364, 9709348326636, 16331909651988, 31948891146732, 54770416120644, 100509779504316, 208751080955844, 388416032284476, 749365894850244, 1414070378301756, 2556878765995204, 2556878765995260, 6726041614128900, 15626498692840700, 23762659088671300, 35168735451235260, 78257359358590020, 186897487211247036, 340813223900632644, 592585414385033916, 1326583294186844484, 2594892903616159356, 4946738730471899844, 8244565422068579772, 13740942370114299844, 13780400058385352252, 13780400058385352308, 14272557426581383244, 14272557426581383300, 21155073391000330684, 21374326697892540932, 22138822441861473292, … (A008892 of Sloane’s OEIS provides links to far more extensive listings). It keeps going like that for at least another thousand terms, and the numbers get so large that their factorizations begin to take noticeable amounts of time. Even so, this aliquot sequence is not in strict

ascending order

. In fact, it is an open question if it’s even possible for an aliquot sequence to be in ascending order.

However, there are aliquot sequences in

descending order

: 135, 105, 87, 33, 15, 9, 4, 3, 1, 0.

In exploring aliquot sequences with a computer, one needs to be careful not just with aliquot sequences of unknown length, but also with those where even the human

operator

already knows how short the

sequence

is. Even a sophisticated

computer algebra system

like

Mathematica

can get stuck on the aliquot sequence for 220 or 284 (

amicable numbers

) if the operator neglects to

program

in the ability to recognize cycles.

Title

examples of aliquot sequences

Canonical name

ExamplesOfAliquotSequences

Date of creation

2013-03-22 16:09:21

Last modified on

2013-03-22 16:09:21

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A25

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