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Perfect Numbers

Perfect numbers are numbers for which sum of all its factors is twice the number.
For example,
6: 1, 2, 3, 6
28: 1, 2, 4, 7, 14, 28
496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
8128: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128

It is not difficult to prove that every even perfect number has the form [(2^p)-1][2^(p-1)], where p is a Mersenne Prime, although there are no known odd perfect numbers, nor there is a proof that no odd perfect numbers exist.

Let sigma(n) denote the sum of all factors of n, including n.
And s(n) be the sum of all factors of n, except n.

So, for a perfect number, s(n)=n, or sigma(n)=2n.

Amicable numbers

Amicable numbers are two numbers (x, y) such that s(x)=y and s(y)=x.
For example,
220 284
1184 1210
2620 2924
5020 5564
6232 6368
10744 10856
12285 14595
17296 18416
Since amicable numbers occur randomly, without certain conditions, there are no theorems for it.

Sociable numbers
Similarly, if there are three numbers (x,y,z) such that s(x)=y, s(y)=z and s(z)=x, then they are called sociable numbers of order 3, or an Aliquot 3 cycle.

So far, we know 165 cycles of order 4, one of order 5, five of order 6, two of order 8, one of order 9, and one of order 28.

The first few Aliquot 4 cycles are
1264460 1547860 1727636 1305184
2115324 3317740 3649556 2797612
2784580 3265940 3707572 3370604
4938136 5753864 5504056 5423384
7169104 7538660 8292568 7520432
18048976 20100368 18914992 19252208
18656380 20522060 28630036 24289964
28158165 29902635 30853845 29971755
46722700 56833172 53718220 59090084
81128632 91314968 96389032 91401368
174277820 205718020 262372988 210967684
209524210 246667790 231439570 230143790
330003580 363003980 399304420 440004764
498215416 506040584 583014136 510137384

The Aliquot 5 cycle is
12496 14288 15472 14536 14264

Aliquot 28 cycle:
14316 19116 31704 47616 83328 177792 295488 629072 589786 294896 358336 418904 366556 274924 275444 243760 376736 381028 285778 152990 122410 97946 48976 45946 22976 22744 19916 17716

Aliquot 6 cycles:
21548919483 23625285957 24825443643 26762383557 25958284443 23816997477
90632826380 101889891700 127527369100 159713440756 129092518924 106246338676
1771417411016 1851936384424 2118923133656 2426887897384 2200652585816 2024477041144
3524434872392 4483305479608 4017343956392 4574630214808 4018261509992 3890837171608
4773123705616 5826394399664 5574013457296 5454772780208 5363145542992 5091331952624

Aliquot 8 cycles:
1095447416 1259477224 1156962296 1330251784 1221976136 1127671864 1245926216 1213138984
1276254780 2299401444 3071310364 2303482780 2629903076 2209210588 2223459332 1697298124

Aliquot 9 cycle:
805984760 1268997640 1803863720 2308845400 3059220620 3367978564 2525983930 2301481286 1611969514

The Aliquot Sequence

The sequence formed by keeping on iterating s(n), starting with a certain value of N, is called the Aliquot sequence. For example,
30 42 54 66 78 90 144 259 45 33 15 9 4 3
60 108 172 136 134 70 74 40 50 43
96 156 236 184 176 196 203 37
48 76 64 63 41
12 16 15 9 4 3
18 21 11
24 36 55 17
120 240 504 1056 1968 3240 7650 14112 32571 27333 12161

Usually, aliquot sequence terminates in a prime.
However, this may not always be the case, as for example, we have, so that,
2195 445 95 25 6 6
which terminates in the perfect number, namely, 6.

So far, it has been conjectured that every aliquot sequence terminates in a prime or an aliquot cycle. However, the following sequences do not.
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, 5148, 5208, 5250, 5352, 5400, 5448, 5736, 5748, 5778, 6396, 6552, 6680, 6822, 6832, 6984, 7044, 7392, 7560, 7890, 7920, 8040, 8154, 8184, 8288, 8352, 8760, 8844, 8904, 9120, 9282, 9336, 9378, 9436, 9462, 9480, 9588, 9684, 9708, 9852

They are called open end aliquot sequences. Computation is going on to check the progress of these open end sequences. These sequences have risen upto atleast 100 digits in size.

The first five, namely, 276, 552, 564, 660, 966 are called Lehmer Five. However, it was called Lehmer Six (along with 840), however the sequence 840 was shown to terminate.

The twelve numbers between 1000 and 2000, (along with 1248 and 1848) are called Godwin Fourteen. As usually, as before, and as expected, the sequences 1248 and 1848 have then been shown to terminate, thus.