Problem 341
Golomb’s self-describing sequence
The Golomb’s self-describing sequence {G(n)} is the only nondecreasing sequence of natural numbers such that n appears exactly G(n) times in the sequence. The values of G(n) for the first few n are
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | … |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| G(n) | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | … |
You are given that G(103) = 86, G(106) = 6137.
You are also given that ΣG(n3) = 153506976 for 1 ≤ n < 103.
Find ΣG(n3) for 1 ≤ n < 106.
戈洛姆的自描述序列
戈洛姆的自描述序列 {G(n)}是唯一一个单调不降且n出现恰好G(n)次的自然数序列。对于前几个n，G(n)的值为
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | … |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| G(n) | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | … |
已知G(103) = 86，G(106) = 6137。
此外，还已知对于1 ≤ n < 103，ΣG(n3) = 153506976。
对于1 ≤ n < 106，求ΣG(n3)。