Sociable Chain/Examples/14,316

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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 $1918$.


Sources