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This is only conjectural, since we don't know that A003416 is complete (cf. A122726). - N. J. A. Sloane, Sep 17 2021

120 sociable numbers of order 4 are known as of Feb. 2003.

142 were known in 2007 (http://amicable.homepage.dk/knwnc4.htm).

201 are known in 2012.

210 were known in April 2013. - Michel Marcus, Nov 10 2013

From Amiram Eldar, Mar 24 2024: (Start)

The terms were found by:

a(1)-a(2) - Kenneth Dudley Fryer in 1965 (Honsberger, 1970; see also A072892)

a(3)-a(7), a(9) - Cohen (1970)

a(8) - Borho (1969)

a(10)-a(13) - independently by Richard David (1972; Devitt et al., 1976, Guy 1977) and Steve C. Root (Beeler et al. 1972)

a(14) - Steve C. Root in 1972

a(15)-a(22) - Flammenkamp (1991)

a(23)-a(24) - Moews and Moews (1991)

a(25)-a(27) - Moews and Moews (1993)

(End)

Comments from David Moews, Sep 17 2021: (Start)

It is possible that there are quite small numbers missing from this sequence. There is no proof that 564 (for example) is missing.

Let s(n) = sigma(n)-n denote the sum of the divisors of n, excluding n itself. The aliquot sequence starting at n is the sequence n, s(n), s(s(n)), s(s(s(n))), ...

Starting at 564, the aliquot sequence continues for at least 3486 steps, reaching a 198-digit number after 3486 iterations of s. In the reverse direction, 563 = 564 - 1 is prime so s(563^2) = 564 and also s(7*316961) = 563^2, s(17*2218709) = 7*316961, etc.; given a strengthened form of the Goldbach conjecture (see Booker, 2018), one can continue iterating s^(-1) indefinitely.

Although it seems unlikely, I don't see any way to be completely certain that the forward aliquot sequence doesn't meet the backwards tree; if it did, 564 would be part of a (very long) aliquot cycle.

Many other numbers below 79750 are in a similar situation (although not 276, because it is not in the image of s).

(Added Sep 18 2021) The smallest uncertain number is 564. All smaller numbers either have known aliquot sequences (all except 276, 306, 396, and 552), are not in the image of s (276, 306, and 552), or are in the image of s but not the image of s^2 (396).

(End)

Called a "sociable" chain.

One of the two aliquot cycles of length greater than 2 that were discovered by Belgian mathematician Paul Poulet (1887-1946) in 1918 (the second is A072890). They were the only known such cycles until 1965 (see A072892). - Amiram Eldar, Mar 24 2024

A k-sociable (or multisociable) cycle of order r consists of r distinct positive integers such that the sum of the aliquot divisors (or proper divisors) of each is equal to k times the next term in the cycle, where k (the multiplicity) is a fixed positive integer.

A183017(n) gives the multiplicity of the cycle with smallest term a(n).

A183018(n) gives the order of the cycle with smallest term a(n).

If examples of two or more multisociable cycles with the same smallest term exist, the smallest term is repeated in this sequence, and corresponding multiplicities listed in order of increasing size in A183017. (No such examples are known. Do any exist?)

If n=product p_i^a_i, d=product p_i^c_i is a unitary divisor of n if each c_i is 0 or a_i.

If n = Product p_i^a_i, d = Product p_i^c_i is an infinitary divisor of n if each c_i has a zero bit in its binary representation everywhere that the corresponding a_i does.

From Amiram Eldar, Mar 25 2023: (Start)

Analogous to A003416 with the sum of the aliquot infinitary divisors function (A126168) instead of A001065.

Only cycles of length greater than 2 are here. Cycles of length 1 correspond to infinitary perfect numbers (A007357), and cycles of length 2 correspond to infinitary amicable pairs (A126169 and A126170).

The corresponding cycles are of lengths 4, 4, 4, 6, 4, 4, 4, 4, 11, 6, 4, 6, 4, 11, 6, 23, 4, 4, 85, 4, 4, 4, 4, 4, 4, ...

It is conjectured that there are no missing terms in the data, but it was not proven. For example, it is not known that the infinitary aliquot sequence that starts at 840 does not reach 840 again (see A361421). (End)

By definition, this is the union of A000396, A259180, and A122726. However, at present A122726 is not known to be complete. There is no proof that 564 (for example) is missing from this sequence. - N. J. A. Sloane, Sep 17 2021

Numbers m for which there exists k>=1 such that s^k(m) = m, where s is A001065.

Conjecture: There are no aliquot cycles containing even numbers and odd numbers simultaneously, i.e., every aliquot cycle either has only even numbers or has only odd numbers.

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).

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.