For 1000 <= n <= 1259, (12336 <= u(n) <= 15466), I get the following distribution of f(n), which seems to support your observation: range of f(n) count [0, 0.050) 37 [0.050, 0.10) 25 [0.10, 0.15) 29 [0.15, 0.20) 22 [0.20, 0.25) 20 [0.25, 0.30) 11 [0.30, 0.35) 12 [0.35, 0.40) 7 [0.40, 0.45) 10 [0.45, 0.50) 1 [0.50, 0.55) 2 [0.55, 0.60) 0 [0.60, 0.65) 0 [0.65, 0.70) 0 [0.70, 0.75) 2 [0.75, 0.80) 6 [0.80, 0.85) 6 [0.85, 0.90) 24 [0.90, 0.95) 21 [0.95, 1.0) 25 Klaus -------------------------------------------------------------------

David Wilson wrote:

I took a look at Ulam(1,2), the Ulam sequence starting with (1, 2) and including every subsequent number which is a unique sum of distinct earlier terms. This is Sloane's A002858.

I took this sequence out quite a ways, and I noticed that after an initial flurry of numbers that are fairly uniform in distribution, the sequence starts to separate out into more or less regular clumps of numbers with a period that seems to be slighly more than 21.6. Between these clumps are spaces containing relatively few numbers.

To see the phenomenon, let u(n) be the nth Ulam(1,2) number, and define

f(n) = u(n) / 21.6 - [ u(n) / 2.16 ]

f(n) is a number on [0, 1) which indicates "u(n) mod 21.6". If we compute f(n) for 1000 <= n <= 1999, (12336 <= u(n) <= 25511), we find the following distribution of f(n):

range of f(n) count [0.00, 0.05) 110 [0.05, 0.10) 101 [0.10, 0.15) 120 [0.15, 0.20) 97 [0.20, 0.25) 90 [0.25, 0.30) 67 [0.30, 0.35) 58 [0.35, 0.40) 40 [0.40, 0.45) 31 [0.45, 0.50) 9 [0.50, 0.55) 4 [0.55, 0.60) 0 [0.60, 0.65) 0 [0.65, 0.70) 2 [0.70, 0.75) 7 [0.75, 0.80) 11 [0.80, 0.85) 22 [0.85, 0.90) 64 [0.90, 0.95) 66 [0.95, 1.00) 101

The values of f(n) are not uniformly distributed, the distribution indicate that u(n) values clump together with clumps occuring with a period of about 21.6.

This all presupposes that my computations of Ulma(1,2) are correct. Can anyone confirm my data?

Has this phenomenon been noted in the literature?