# Sociable Chain/Examples/14,316

## Example of Sociable Chain

The longest known sociable chain, at time of writing ($9$th March $2017$), is of order $28$.

Its smallest element is $14 \, 316$.

## Proof

Let $\map s n$ denote the aliquot sum of $n$.

By definition:

$\map s n = \map \sigma n - n$

where $\map \sigma n$ denotes the $\sigma$ function.

Thus:

 $\displaystyle \map s {14 \, 316}$ $=$ $\displaystyle \map \sigma {14 \, 316} - 14 \, 316$ $\displaystyle$ $=$ $\displaystyle 33 \, 432 - 14 \, 316$ $\sigma$ of $14 \, 316$ $\displaystyle$ $=$ $\displaystyle 19 \, 116$

 $\displaystyle \map s {19 \, 116}$ $=$ $\displaystyle \map \sigma {19 \, 116} - 19 \, 116$ $\displaystyle$ $=$ $\displaystyle 50 \, 820 - 19 \, 116$ $\sigma$ of $19 \, 116$ $\displaystyle$ $=$ $\displaystyle 31 \, 704$

 $\displaystyle \map s {31 \, 704}$ $=$ $\displaystyle \map \sigma {31 \, 704} - 31 \, 704$ $\displaystyle$ $=$ $\displaystyle 79 \, 320 - 31 \, 704$ $\sigma$ of $31 \, 704$ $\displaystyle$ $=$ $\displaystyle 47 \, 616$

 $\displaystyle \map s {47 \, 616}$ $=$ $\displaystyle \map \sigma {47 \, 616} - 47 \, 616$ $\displaystyle$ $=$ $\displaystyle 130 \, 994 - 47 \, 616$ $\sigma$ of $47 \, 616$ $\displaystyle$ $=$ $\displaystyle 83 \, 328$

 $\displaystyle \map s {83 \, 328}$ $=$ $\displaystyle \map \sigma {83 \, 328} - 83 \, 328$ $\displaystyle$ $=$ $\displaystyle 261 \, 120 - 83 \, 328$ $\sigma$ of $83 \, 328$ $\displaystyle$ $=$ $\displaystyle 177 \, 792$

 $\displaystyle \map s {177 \, 792}$ $=$ $\displaystyle \map \sigma {177 \, 792} - 177 \, 792$ $\displaystyle$ $=$ $\displaystyle 473 \, 280 - 177 \, 792$ $\sigma$ of $177 \, 792$ $\displaystyle$ $=$ $\displaystyle 295 \, 488$

 $\displaystyle \map s {295 \, 488}$ $=$ $\displaystyle \map \sigma {295 \, 488} - 295 \, 488$ $\displaystyle$ $=$ $\displaystyle 924 \, 560 - 295 \, 488$ $\sigma$ of $295 \, 488$ $\displaystyle$ $=$ $\displaystyle 629 \, 072$

 $\displaystyle \map s {629 \, 072}$ $=$ $\displaystyle \map \sigma {629 \, 072} - 629 \, 072$ $\displaystyle$ $=$ $\displaystyle 1 \, 218 \, 858 - 629 \, 072$ $\sigma$ of $629 \, 072$ $\displaystyle$ $=$ $\displaystyle 589 \, 786$

 $\displaystyle \map s {589 \, 786}$ $=$ $\displaystyle \map \sigma {589 \, 786} - 589 \, 786$ $\displaystyle$ $=$ $\displaystyle 884 \, 682 - 589 \, 786$ $\sigma$ of $589 \, 786$ $\displaystyle$ $=$ $\displaystyle 294 \, 896$

 $\displaystyle \map s {294 \, 896}$ $=$ $\displaystyle \map \sigma {294 \, 896} - 294 \, 896$ $\displaystyle$ $=$ $\displaystyle 653 \, 232 - 294 \, 896$ $\sigma$ of $294 \, 896$ $\displaystyle$ $=$ $\displaystyle 358 \, 336$

 $\displaystyle \map s {358 \, 336}$ $=$ $\displaystyle \map \sigma {358 \, 336} - 358 \, 336$ $\displaystyle$ $=$ $\displaystyle 777 \, 240 - 358 \, 336$ $\sigma$ of $358 \, 336$ $\displaystyle$ $=$ $\displaystyle 418 \, 904$

 $\displaystyle \map s {418 \, 904}$ $=$ $\displaystyle \map \sigma {418 \, 904} - 418 \, 904$ $\displaystyle$ $=$ $\displaystyle 785 \, 460 - 418 \, 904$ $\sigma$ of $418 \, 904$ $\displaystyle$ $=$ $\displaystyle 366 \, 556$

 $\displaystyle \map s {366 \, 556}$ $=$ $\displaystyle \map \sigma {366 \, 556} - 366 \, 556$ $\displaystyle$ $=$ $\displaystyle 641 \, 480 - 366 \, 556$ $\sigma$ of $366 \, 556$ $\displaystyle$ $=$ $\displaystyle 274 \, 924$

 $\displaystyle \map s {274 \, 924}$ $=$ $\displaystyle \map \sigma {274 \, 924} - 274 \, 924$ $\displaystyle$ $=$ $\displaystyle 550 \, 368 - 274 \, 924$ $\sigma$ of $274 \, 924$ $\displaystyle$ $=$ $\displaystyle 275 \, 444$

 $\displaystyle \map s {275 \, 444}$ $=$ $\displaystyle \map \sigma {275 \, 444} - 275 \, 444$ $\displaystyle$ $=$ $\displaystyle 519 \, 204 - 275 \, 444$ $\sigma$ of $275 \, 444$ $\displaystyle$ $=$ $\displaystyle 243 \, 760$

 $\displaystyle \map s {243 \, 760}$ $=$ $\displaystyle \map \sigma {243 \, 760} - 243 \, 760$ $\displaystyle$ $=$ $\displaystyle 620 \, 496 - 243 \, 760$ $\sigma$ of $243 \, 760$ $\displaystyle$ $=$ $\displaystyle 376 \, 736$

 $\displaystyle \map s {376 \, 736}$ $=$ $\displaystyle \map \sigma {376 \, 736} - 376 \, 736$ $\displaystyle$ $=$ $\displaystyle 757 \, 764 - 376 \, 736$ $\sigma$ of $376 \, 736$ $\displaystyle$ $=$ $\displaystyle 381 \, 028$

 $\displaystyle \map s {381 \, 028}$ $=$ $\displaystyle \map \sigma {381 \, 028} - 381 \, 028$ $\displaystyle$ $=$ $\displaystyle 666 \, 806 - 381 \, 028$ $\sigma$ of $381 \, 028$ $\displaystyle$ $=$ $\displaystyle 285 \, 778$

 $\displaystyle \map s {285 \, 778}$ $=$ $\displaystyle \map \sigma {285 \, 778} - 285 \, 778$ $\displaystyle$ $=$ $\displaystyle 438 \, 768 - 285 \, 778$ $\sigma$ of $285 \, 778$ $\displaystyle$ $=$ $\displaystyle 152 \, 990$

 $\displaystyle \map s {152 \, 990}$ $=$ $\displaystyle \map \sigma {152 \, 990} - 152 \, 990$ $\displaystyle$ $=$ $\displaystyle 275 \, 400 - 152 \, 990$ $\sigma$ of $152 \, 990$ $\displaystyle$ $=$ $\displaystyle 122 \, 410$

 $\displaystyle \map s {122 \, 410}$ $=$ $\displaystyle \map \sigma {122 \, 410} - 122 \, 410$ $\displaystyle$ $=$ $\displaystyle 220 \, 356 - 122 \, 410$ $\sigma$ of $122 \, 410$ $\displaystyle$ $=$ $\displaystyle 97 \, 946$

 $\displaystyle \map s {97 \, 946}$ $=$ $\displaystyle \map \sigma {97 \, 946} - 97 \, 946$ $\displaystyle$ $=$ $\displaystyle 146 \, 922 - 97 \, 946$ $\sigma$ of $97 \, 946$ $\displaystyle$ $=$ $\displaystyle 48 \, 976$

 $\displaystyle \map s {48 \, 976}$ $=$ $\displaystyle \map \sigma {48 \, 976} - 48 \, 976$ $\displaystyle$ $=$ $\displaystyle 94 \, 922 - 48 \, 976$ $\sigma$ of $48 \, 976$ $\displaystyle$ $=$ $\displaystyle 45 \, 946$

 $\displaystyle \map s {45 \, 946}$ $=$ $\displaystyle \map \sigma {45 \, 946} - 45 \, 946$ $\displaystyle$ $=$ $\displaystyle 68 \, 922 - 45 \, 946$ $\sigma$ of $45 \, 946$ $\displaystyle$ $=$ $\displaystyle 22 \, 976$

 $\displaystyle \map s {22 \, 976}$ $=$ $\displaystyle \map \sigma {22 \, 976} - 22 \, 976$ $\displaystyle$ $=$ $\displaystyle 45 \, 720 - 22 \, 976$ $\sigma$ of $22 \, 976$ $\displaystyle$ $=$ $\displaystyle 22 \, 744$

 $\displaystyle \map s {22 \, 744}$ $=$ $\displaystyle \map \sigma {22 \, 744} - 22 \, 744$ $\displaystyle$ $=$ $\displaystyle 42 \, 660 - 22 \, 744$ $\sigma$ of $22 \, 744$ $\displaystyle$ $=$ $\displaystyle 19 \, 916$

 $\displaystyle \map s {19 \, 916}$ $=$ $\displaystyle \map \sigma {19 \, 916} - 19 \, 916$ $\displaystyle$ $=$ $\displaystyle 37 \, 632 - 19 \, 916$ $\sigma$ of $19 \, 916$ $\displaystyle$ $=$ $\displaystyle 17 \, 716$

 $\displaystyle \map s {17 \, 716}$ $=$ $\displaystyle \map \sigma {17 \, 716} - 17 \, 716$ $\displaystyle$ $=$ $\displaystyle 32 \, 032 - 17 \, 716$ $\sigma$ of $17 \, 716$ $\displaystyle$ $=$ $\displaystyle 14 \, 316$

It is interesting to list the prime decomposition of each of the terms in the chain:

 $\displaystyle 14 \, 316$ $=$ $\displaystyle 2^2 \times 3 \times 1193$ $\displaystyle 19 \, 116$ $=$ $\displaystyle 2^2 \times 3^4 \times 59$ $\displaystyle 31 \, 704$ $=$ $\displaystyle 2^3 \times 3 \times 1321$ $\displaystyle 47 \, 616$ $=$ $\displaystyle 2^9 \times 3 \times 31$ $\displaystyle 83 \, 328$ $=$ $\displaystyle 2^7 \times 3 \times 7 \times 31$ $\displaystyle 177 \, 792$ $=$ $\displaystyle 2^7 \times 3 \times 7 \times 463$ $\displaystyle 295 \, 488$ $=$ $\displaystyle 2^6 \times 3^5 \times 19$ $\displaystyle 629 \, 072$ $=$ $\displaystyle 2^4 \times 39 \, 317$ $\displaystyle 589 \, 786$ $=$ $\displaystyle 2 \times 294 \, 893$ $\displaystyle 294 \, 896$ $=$ $\displaystyle 2^4 \times 7 \times 2633$ $\displaystyle 358 \, 336$ $=$ $\displaystyle 2^6 \times 11 \times 509$ $\displaystyle 418 \, 904$ $=$ $\displaystyle 2^3 \times 52 \, 363$ $\displaystyle 366 \, 556$ $=$ $\displaystyle 2^2 \times 91 \, 639$ $\displaystyle 274 \, 924$ $=$ $\displaystyle 2^2 \times 13 \times 17 \times 311$ $\displaystyle 275 \, 444$ $=$ $\displaystyle 2^2 \times 13 \times 5297$ $\displaystyle 243 \, 760$ $=$ $\displaystyle 2^4 \times 5 \times 11 \times 277$ $\displaystyle 376 \, 736$ $=$ $\displaystyle 2^5 \times 61 \times 193$ $\displaystyle 381 \, 028$ $=$ $\displaystyle 2^2 \times 95 \, 257$ $\displaystyle 285 \, 778$ $=$ $\displaystyle 2 \times 43 \times 3323$ $\displaystyle 152 \, 990$ $=$ $\displaystyle 2 \times 5 \times 15 \, 299$ $\displaystyle 122 \, 410$ $=$ $\displaystyle 2 \times 5 \times 12 \, 241$ $\displaystyle 97 \, 946$ $=$ $\displaystyle 2 \times 48 \, 973$ $\displaystyle 48 \, 976$ $=$ $\displaystyle 2^4 \times 3061$ $\displaystyle 45 \, 946$ $=$ $\displaystyle 2 \times 22 \, 973$ $\displaystyle 22 \, 976$ $=$ $\displaystyle 2^6 \times 359$ $\displaystyle 22 \, 744$ $=$ $\displaystyle 2^3 \times 2843$ $\displaystyle 19 \, 916$ $=$ $\displaystyle 2^2 \times 13 \times 383$ $\displaystyle 17 \, 716$ $=$ $\displaystyle 2^2 \times 43 \times 103$

$\blacksquare$

## Historical Note

The sociable chain of $14 \, 316$ was discovered by Paul Poulet.

He included a reference to it in a paper he published in $1918$.