Sociable number Contents Example List of known sociable numbers Searching for sociable numbers Conjecture of the sum of sociable number cycles References External links Navigation menuProofWiki: Catalan-Dickson Conjecture"Amicable numbers and their distribution"10.1090/S0025-5718-1970-0271005-80025-5718Amicable pairs list10.13140/RG.2.1.1233.8640A list of known sociable numbersExtensive tables of perfect, amicable and sociable numbers"Sociable numbers"ee
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Divisor functionInteger sequencesNumber theory
aliquot sumsamicable numbersperfect numbersBelgianmathematicianPaul Pouletproper factorsperiodperfect numberproper divisorsamicable numbersprimealiquot sequencealiquot sequencedirected graphCyclesperfect numbersamicable pairsOEIS
Sociable numbers are numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number. They are generalizations of the concepts of amicable numbers and perfect numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918.[1] In a set of sociable numbers, each number is the sum of the proper factors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to 5×107displaystyle 5times 10^7 as of 1970 [2].
It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
Contents
1 Example
2 List of known sociable numbers
3 Searching for sociable numbers
4 Conjecture of the sum of sociable number cycles
5 References
6 External links
Example
An example with period 4:
- The sum of the proper divisors of 1264460displaystyle 1264460 (=22⋅5⋅17⋅3719displaystyle =2^2cdot 5cdot 17cdot 3719) is:
- 1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860
- The sum of the proper divisors of 1547860displaystyle 1547860 (=22⋅5⋅193⋅401displaystyle =2^2cdot 5cdot 193cdot 401) is:
- 1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636
- The sum of the proper divisors of 1727636displaystyle 1727636 (=22⋅521⋅829displaystyle =2^2cdot 521cdot 829) is:
- 1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184
- The sum of the proper divisors of 1305184displaystyle 1305184 (=25⋅40787displaystyle =2^5cdot 40787) is:
- 1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
List of known sociable numbers
The following categorizes all known sociable numbers as of July 2018 by the length of the corresponding aliquot sequence:
| Sequence length | Number of known sequences |
|---|---|
| 1 (Perfect number) | 50 |
| 2 (Amicable number) | 1222207191[3] |
| 4 | 5398 |
| 5 | 1 |
| 6 | 5 |
| 8 | 4 |
| 9 | 1 |
| 28 | 1 |
It is conjectured that if n = 3 mod 4, then there are no such sequence with length n.
The smallest number of the only known 28-cycle is 14316.
Searching for sociable numbers
The aliquot sequence can be represented as a directed graph, Gn,sdisplaystyle G_n,s, for a given integer ndisplaystyle n, where s(k)displaystyle s(k) denotes the
sum of the proper divisors of kdisplaystyle k.[4]Cycles in Gn,sdisplaystyle G_n,s represent sociable numbers within the interval [1,n]displaystyle [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
![[1,n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c79af450e22e8fd23f28e6be4cb23a47b24c1ba)
Conjecture of the sum of sociable number cycles
As the number of sociable number cycles with length greater than 2 approaches infinity, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%.(sequence A292217 in the OEIS).
References
^ P. Poulet, #4865, L'Intermédiaire des Mathématiciens 25 (1918), pp. 100–101. (The full text can be found at ProofWiki: Catalan-Dickson Conjecture.)
^ Bratley, Paul; Lunnon, Fred; McKay, John (1970). "Amicable numbers and their distribution". Mathematics of Computation. 24 (110): 431–432. doi:10.1090/S0025-5718-1970-0271005-8. ISSN 0025-5718..mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
^ Sergei Chernykh Amicable pairs list
^ Rocha, Rodrigo Caetano; Thatte, Bhalchandra (2015), Distributed cycle detection in large-scale sparse graphs, Simpósio Brasileiro de Pesquisa Operacional (SBPO), doi:10.13140/RG.2.1.1233.8640
- H. Cohen, On amicable and sociable numbers, Math. Comp. 24 (1970), pp. 423–429
External links
- A list of known sociable numbers
- Extensive tables of perfect, amicable and sociable numbers
- Weisstein, Eric W. "Sociable numbers". MathWorld.
A003416 (smallest sociable number from each cycle) and A122726 (all sociable numbers) in OEIS
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