Sylvester's sequence
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Sylvester's sequence consists of the coprime denominators of an Egyptian fraction that adds up to 1. The first few terms of the sequence are
2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 (sequence in OEIS)
The terms of the sequence can be calculated in two different ways:
[e_n = 1 + \prod_^ e_i = 1 + (e_)^2 - e_] with [e_0=2]
Putting 1s as numerators to these numbers and adding up the fractions results in a sum that converges to 1, as the following table shows:
| 2 | 1/2 ... | 1/2 | 0.5 |
| 3 | ... + 1/3 ... | 5/6 | 0.833... |
| 7 | ... + 1/7 ... | 41/42 | 0.976190476190476... |
| 43 | ... + 1/43 ... | 1805/1806 | 0.99944629014396456257... |
| 1807 | ... + 1/1807 ... | 3263441/3263442 | 0.99999969357506583540... |
| 3263443 | ... + 1/3263443 ... | 10650056950805/10650056950806 | 0.99999999999990610379... |
| 10650056950807 | ... + 1/10650056950807 + ... | 113423713055421844361000441 113423713055421844361000442 | 0.99999999999999999999... |
Or to put it algebraically,
[\lim_ \sum_^n } = 1]
Sylvester's sequence is useful in the rational approximation of irrational numbers using a greedy algorithm.
Although it is obvious that all the terms of Sylvester's sequence are coprime, it's not known if they are all squarefree (all the known terms are).
In the set of solutions to Znám's problem for a given length k, it is likely that at least one of the solutions will contain the first k - 2 numbers of Sylvester's sequence.
Some effort has been expended in an attempt to factor these numbers. They are known to be composite for [6\le n\le18].
| n | Factors of en |
|---|---|
| 6 | 547 × 607 × 1033 × 31051 |
| 7 | 29881 × 67003 × 9119521 × 6212157481 |
| 8 | 5295435634831 × 31401519357481261 × 77366930214021991992277 |
| 9 | 181 × 1987 × 112374829138729 × 114152531605972711 × P68 |
| 10 | 2287 × 2271427 × 21430986826194127130578627950810640891005487 × P156 |
| 11 | 73 × C416 |
| 12 | 2589377038614498251653 × 2872413602289671035947763837 × C785 |
| 13 | 52387 × 5020387 × 5783021473 × 401472621488821859737 × C1626 |
| 14 | 13999 × 74203 × 9638659 × 57218683 × 10861631274478494529 × C3293 |
| 15 | 17881 × 97822786011310111 × C6649 |
| 16 | 128551 × C13336 |
| 17 | 635263 × 1286773 × 21269959 × C26661 |
| 18 | 139263586549 × C53349 |
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