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:

$\map s n = \map {\sigma_1} n - n$

where $\map {\sigma_1} n$ denotes the divisor sum function.


Thus:

\(\ds \map s {14 \, 316}\) \(=\) \(\ds \map {\sigma_1} {14 \, 316} - 14 \, 316\)
\(\ds \) \(=\) \(\ds 33 \, 432 - 14 \, 316\) $\sigma_1$ of $14 \, 316$
\(\ds \) \(=\) \(\ds 19 \, 116\)


\(\ds \map s {19 \, 116}\) \(=\) \(\ds \map {\sigma_1} {19 \, 116} - 19 \, 116\)
\(\ds \) \(=\) \(\ds 50 \, 820 - 19 \, 116\) $\sigma_1$ of $19 \, 116$
\(\ds \) \(=\) \(\ds 31 \, 704\)


\(\ds \map s {31 \, 704}\) \(=\) \(\ds \map {\sigma_1} {31 \, 704} - 31 \, 704\)
\(\ds \) \(=\) \(\ds 79 \, 320 - 31 \, 704\) $\sigma_1$ of $31 \, 704$
\(\ds \) \(=\) \(\ds 47 \, 616\)


\(\ds \map s {47 \, 616}\) \(=\) \(\ds \map {\sigma_1} {47 \, 616} - 47 \, 616\)
\(\ds \) \(=\) \(\ds 130 \, 994 - 47 \, 616\) $\sigma_1$ of $47 \, 616$
\(\ds \) \(=\) \(\ds 83 \, 328\)


\(\ds \map s {83 \, 328}\) \(=\) \(\ds \map {\sigma_1} {83 \, 328} - 83 \, 328\)
\(\ds \) \(=\) \(\ds 261 \, 120 - 83 \, 328\) $\sigma_1$ of $83 \, 328$
\(\ds \) \(=\) \(\ds 177 \, 792\)


\(\ds \map s {177 \, 792}\) \(=\) \(\ds \map {\sigma_1} {177 \, 792} - 177 \, 792\)
\(\ds \) \(=\) \(\ds 473 \, 280 - 177 \, 792\) $\sigma_1$ of $177 \, 792$
\(\ds \) \(=\) \(\ds 295 \, 488\)


\(\ds \map s {295 \, 488}\) \(=\) \(\ds \map {\sigma_1} {295 \, 488} - 295 \, 488\)
\(\ds \) \(=\) \(\ds 924 \, 560 - 295 \, 488\) $\sigma_1$ of $295 \, 488$
\(\ds \) \(=\) \(\ds 629 \, 072\)


\(\ds \map s {629 \, 072}\) \(=\) \(\ds \map {\sigma_1} {629 \, 072} - 629 \, 072\)
\(\ds \) \(=\) \(\ds 1 \, 218 \, 858 - 629 \, 072\) $\sigma_1$ of $629 \, 072$
\(\ds \) \(=\) \(\ds 589 \, 786\)


\(\ds \map s {589 \, 786}\) \(=\) \(\ds \map {\sigma_1} {589 \, 786} - 589 \, 786\)
\(\ds \) \(=\) \(\ds 884 \, 682 - 589 \, 786\) $\sigma_1$ of $589 \, 786$
\(\ds \) \(=\) \(\ds 294 \, 896\)


\(\ds \map s {294 \, 896}\) \(=\) \(\ds \map {\sigma_1} {294 \, 896} - 294 \, 896\)
\(\ds \) \(=\) \(\ds 653 \, 232 - 294 \, 896\) $\sigma_1$ of $294 \, 896$
\(\ds \) \(=\) \(\ds 358 \, 336\)


\(\ds \map s {358 \, 336}\) \(=\) \(\ds \map {\sigma_1} {358 \, 336} - 358 \, 336\)
\(\ds \) \(=\) \(\ds 777 \, 240 - 358 \, 336\) $\sigma_1$ of $358 \, 336$
\(\ds \) \(=\) \(\ds 418 \, 904\)


\(\ds \map s {418 \, 904}\) \(=\) \(\ds \map {\sigma_1} {418 \, 904} - 418 \, 904\)
\(\ds \) \(=\) \(\ds 785 \, 460 - 418 \, 904\) $\sigma_1$ of $418 \, 904$
\(\ds \) \(=\) \(\ds 366 \, 556\)


\(\ds \map s {366 \, 556}\) \(=\) \(\ds \map {\sigma_1} {366 \, 556} - 366 \, 556\)
\(\ds \) \(=\) \(\ds 641 \, 480 - 366 \, 556\) $\sigma_1$ of $366 \, 556$
\(\ds \) \(=\) \(\ds 274 \, 924\)


\(\ds \map s {274 \, 924}\) \(=\) \(\ds \map {\sigma_1} {274 \, 924} - 274 \, 924\)
\(\ds \) \(=\) \(\ds 550 \, 368 - 274 \, 924\) $\sigma_1$ of $274 \, 924$
\(\ds \) \(=\) \(\ds 275 \, 444\)


\(\ds \map s {275 \, 444}\) \(=\) \(\ds \map {\sigma_1} {275 \, 444} - 275 \, 444\)
\(\ds \) \(=\) \(\ds 519 \, 204 - 275 \, 444\) $\sigma_1$ of $275 \, 444$
\(\ds \) \(=\) \(\ds 243 \, 760\)


\(\ds \map s {243 \, 760}\) \(=\) \(\ds \map {\sigma_1} {243 \, 760} - 243 \, 760\)
\(\ds \) \(=\) \(\ds 620 \, 496 - 243 \, 760\) $\sigma_1$ of $243 \, 760$
\(\ds \) \(=\) \(\ds 376 \, 736\)


\(\ds \map s {376 \, 736}\) \(=\) \(\ds \map {\sigma_1} {376 \, 736} - 376 \, 736\)
\(\ds \) \(=\) \(\ds 757 \, 764 - 376 \, 736\) $\sigma_1$ of $376 \, 736$
\(\ds \) \(=\) \(\ds 381 \, 028\)


\(\ds \map s {381 \, 028}\) \(=\) \(\ds \map {\sigma_1} {381 \, 028} - 381 \, 028\)
\(\ds \) \(=\) \(\ds 666 \, 806 - 381 \, 028\) $\sigma_1$ of $381 \, 028$
\(\ds \) \(=\) \(\ds 285 \, 778\)


\(\ds \map s {285 \, 778}\) \(=\) \(\ds \map {\sigma_1} {285 \, 778} - 285 \, 778\)
\(\ds \) \(=\) \(\ds 438 \, 768 - 285 \, 778\) $\sigma_1$ of $285 \, 778$
\(\ds \) \(=\) \(\ds 152 \, 990\)


\(\ds \map s {152 \, 990}\) \(=\) \(\ds \map {\sigma_1} {152 \, 990} - 152 \, 990\)
\(\ds \) \(=\) \(\ds 275 \, 400 - 152 \, 990\) $\sigma_1$ of $152 \, 990$
\(\ds \) \(=\) \(\ds 122 \, 410\)


\(\ds \map s {122 \, 410}\) \(=\) \(\ds \map {\sigma_1} {122 \, 410} - 122 \, 410\)
\(\ds \) \(=\) \(\ds 220 \, 356 - 122 \, 410\) $\sigma_1$ of $122 \, 410$
\(\ds \) \(=\) \(\ds 97 \, 946\)


\(\ds \map s {97 \, 946}\) \(=\) \(\ds \map {\sigma_1} {97 \, 946} - 97 \, 946\)
\(\ds \) \(=\) \(\ds 146 \, 922 - 97 \, 946\) $\sigma_1$ of $97 \, 946$
\(\ds \) \(=\) \(\ds 48 \, 976\)


\(\ds \map s {48 \, 976}\) \(=\) \(\ds \map {\sigma_1} {48 \, 976} - 48 \, 976\)
\(\ds \) \(=\) \(\ds 94 \, 922 - 48 \, 976\) $\sigma_1$ of $48 \, 976$
\(\ds \) \(=\) \(\ds 45 \, 946\)


\(\ds \map s {45 \, 946}\) \(=\) \(\ds \map {\sigma_1} {45 \, 946} - 45 \, 946\)
\(\ds \) \(=\) \(\ds 68 \, 922 - 45 \, 946\) $\sigma_1$ of $45 \, 946$
\(\ds \) \(=\) \(\ds 22 \, 976\)


\(\ds \map s {22 \, 976}\) \(=\) \(\ds \map {\sigma_1} {22 \, 976} - 22 \, 976\)
\(\ds \) \(=\) \(\ds 45 \, 720 - 22 \, 976\) $\sigma_1$ of $22 \, 976$
\(\ds \) \(=\) \(\ds 22 \, 744\)


\(\ds \map s {22 \, 744}\) \(=\) \(\ds \map {\sigma_1} {22 \, 744} - 22 \, 744\)
\(\ds \) \(=\) \(\ds 42 \, 660 - 22 \, 744\) $\sigma_1$ of $22 \, 744$
\(\ds \) \(=\) \(\ds 19 \, 916\)


\(\ds \map s {19 \, 916}\) \(=\) \(\ds \map {\sigma_1} {19 \, 916} - 19 \, 916\)
\(\ds \) \(=\) \(\ds 37 \, 632 - 19 \, 916\) $\sigma_1$ of $19 \, 916$
\(\ds \) \(=\) \(\ds 17 \, 716\)


\(\ds \map s {17 \, 716}\) \(=\) \(\ds \map {\sigma_1} {17 \, 716} - 17 \, 716\)
\(\ds \) \(=\) \(\ds 32 \, 032 - 17 \, 716\) $\sigma_1$ of $17 \, 716$
\(\ds \) \(=\) \(\ds 14 \, 316\)


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

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

$\blacksquare$


Also see


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

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