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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $14,316$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $14,316$
- Weisstein, Eric W. "Sociable Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SociableNumbers.html