At 12:43 PM 2/11/2003, Edwin Clark wrote:

> Do you still have the sequence up to n = 5 million? If so I would

be interested in knowing whether or not the frequency of i such
that u(i) = u(i-1) + 2 continues to hold up. Gaps between successive
such i are quite small in my limited data (up to n = 2000).

Also u(i) = u(i-1) + 4 only occurs in my data for i = 18 and 28.
And u(i) = u(i-1) + 13 only occurs once (at i = 53) in my list.

Here are the small gaps for the 370,468 Ulam numbers <= 5,000,000

gap    freq   percent

1       4   0.0%
2 136930 37.0%
3   50710 13.7%
4       2   0.0%
5    4057   1.1%
6       0   0.0%
7     806   0.2%
8    4524   1.2%
9       3   0.0%
10     373   0.1%
11       0   0.0%
12   18417   5.0%
13       1   0.0%
14       0   0.0%
15   12557   3.4%
16       0   0.0%
17   31669   8.5%
18       0   0.0%
19     750   0.2%
20   28652   7.7%
21       0   0.0%
22   14623   3.9%
23       0   0.0%
24     748   0.2%
25   20120   5.4%

Gap of 1 occurs only once, 4 only twice, 13 only once, and 9 only 3 times. Gaps of 6, 11, 13, 14, 18, 21, 23 have not occurred.