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.