Problems & Puzzles: Puzzles

Puzzle 215. The Ulam Numbers

For sure you know already what the Ulam numbers are. If not, please read the following articles: 1, 2.

Let's ask here only for the Ulam sequence of numbers starting with 1 & 2.

1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,97,99,102,106,
114,126,131,138,145,148,155,175,177,180,182,189,197,206,209,219,221,236,238,
241,243,253,258,260,273,282,309,316,319,324,339, ...

It has been reported by P. Muller (1966) - who calculated the first 20000 terms, that "more than 60% of these terms differed from another by exactly 2" (UPiNT, R. K. Guy, C4, P. 109)

In a more reduced list of these terms, produced by my own, I have observed that - despite the evidence reported by Muller, the only three consecutive even numbers are 4, 6 & 8.

Questions:

1) Can you find another triplet of consecutive even numbers or to show that they can not exist?

2) Can you produce and share a txt file the first 10^n terms of this sequence, for n as large as you can? (Maybe some readers will find interesting this file in order to study the so many interesting numeric mysteries of this apparently simple sequence)

Solution:

The the asked triplets are impossible was proved by Jon Perry, Luke Pebody, Jud McCranie, Jon Wharf, J. K. Andersen & John Bowker.

Jud McCranie has sent his collection of Ulam numbers as a file which size is 7 MB and can be sent to you on request.

The proof is as simple as this:

"Suppose n,n+2 are in the sequence. Then n+4=(n+2)+2=n+4 cannot be in the sequence" (J. Perry)

***

Some interesting additional observations came for Jon Wharf:

Checking the first 50000 Ulam gaps (Sloanes A072832), I observe the following:

The most popular gaps, each over 1 percent of the total, are:

Gap size     Percent
2              36.896
3              13.678
5              1.236
8              1.204
12             5.034
15             3.448
17             8.326
20             7.66
22             3.916
25             5.342
34             1.224
39             2.212
42             3.26

This is part of a strong pattern of preferred (and spurned) gap sizes. These values are surprising consistent from one batch of 10000 to the next.

The largest gap is 315, occurring once after U18857.
Next largest is 262, occurring once after U4952.
Next largest is 218, occurring six times.
Gaps of size 1 occur only four times - the last after U15 (=47)
Gaps of size 4 occur only twice - the last after U28 (=106)
There are no gaps size 6. Why on earth not?

Easily the most popular multiple-consecutive gap is 22, then 17 and 39. Starting at U43634 (=589633), the next 6 gaps are all of 22. There could perhaps be much larger runs of 22-gaps; the smallest multiple of 22 which is Ulam is 2090 = U223.

The only consecutive gaps of 5 are at U21-U24: 72, 77, 82, 87. After this gaps of 5 appear singly.
The only consecutive gaps of 7 are at U31-U33: 131, 138, 145. After this the only repeating gaps are of 17, 22 or 39.

23 is the first number excluded from the sequence because it is not the sum of 2 members of the sequence. Perhaps these Ulam-unreachable are a study in themselves? The UnUlam sequence starts:
23, 25, 33, 35, 43, 45, 67, 92, 94, 96, 111, 121, 136, 143, 160, 165, 170, 172, 187, 194, 204, 226, 231, 248, 265
(and I see from Sloane's A033629 that Jud McCranie has been here too...)
There are far more UnUlams than Ulams. By U500 you have passed UnU961. Ulams tend to get sparser; UnUlams denser

...

The next few "gaps that never occur", after 6, are 11, 14, 16, 18, 21, 23, 26, 28, 31, 33, 35 and 36 (I just wanted to include that consecutive pair). Additionally there are only 3 gaps size 9, last after U53 = 273 and only 1 size 13 after U52 = 260.

Gaps of 32 provide an opposite case; they are rare, the first being after U2841, but seem to keep appearing intermittently: after U19979, U25434, U31190, U37827

***

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