Ulam Numbers

I've been encountering these numbers for a while now, ever since making use of the website NUMBERS APLENTY which categorises numbers in a wide variety of ways and one of those categories is that of the Ulam number. Yesterday for example I was 25384 days old and 25384 is an Ulam number defined by the site as:

The Ulam sequence is defined by \(U_1=1, U_2=2 \) and, for \( k>2, U_k \) is the smallest integer that can be written in exactly one way as \(U_i+U_j \) with \( i<j<k \). The members of the Ulam sequence are called Ulam numbers. 

For example, after the first 4 terms which are trivially 1, 2, 3 and 4, the value of \(U_5 \) cannot be 5, since \(5=U_1+U_4=U_2+U_3 \), but it is 6, which can be obtained only as  \( U_2+U_4\)  (not as \(U_3+U_3 \) because the terms added must be distinct).

The problem with the Online Encyclopaedia of Integer Sequences (OEIS) is that it will not flag large numbers like 25384 as Ulam numbers because the numbers in the Ulam sequence (OEIS A002858) are only listed up to 339.

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

There is a list of the first 10,000 Ulam numbers in the links but these numbers do not show up in a search of the OEIS. That's why sites like NUMBERS APLENTY are so useful as a supplement to the OEIS. The site always provides a link to the relevant sequence in the OEIS as well as a Mathworld and Wikipedia link.

So what's special about the Ulam numbers? To begin with they are not distributed uniformly. As the NUMBERS APLENTY website states:

Ulam numbers are not distributed uniformly, but their density has peaks at an average distance of 21.6 (see Figure 1):

A plot of the Ulam numbers up to 1082, arranged line by line in a square 108×108. 

The plot shows the peaks in their density that occur with a frequency roughly equal to 108/5=21.6.

Wikipedia claims that the Ulam numbers have a density of approximately 0.07398 or 7.4%. It also states that:

There are infinitely many Ulam numbers. For, after the first n numbers in the sequence have already been determined, it is always possible to extend the sequence by one more element: \( U_{n − 1} + U_n \) is uniquely represented as a sum of two of the first n numbers, and there may be other smaller numbers that are also uniquely represented in this way, so the next element can be chosen as the smallest of these uniquely representable numbers.

Wikipedia goes on to say that:

The idea can be generalised as (u, v)-Ulam numbers by selecting different starting values (u, v). A sequence of (u, v)-Ulam numbers is regular if the sequence of differences between consecutive numbers in the sequence is eventually periodic. When v is an odd number greater than three, the (2, v)-Ulam numbers are regular. When v is congruent to 1 (mod 4) and at least five, the (4, v)-Ulam numbers are again regular. However, the Ulam numbers themselves do not appear to be regular. 

A sequence of numbers is said to be s-additive if each number in the sequence, after the initial 2s terms of the sequence, has exactly s representations as a sum of two previous numbers. Thus, the Ulam numbers and the (u, v)-Ulam numbers are 1-additive sequences. 

If a sequence is formed by appending the largest number with a unique representation as a sum of two earlier numbers, instead of appending the smallest uniquely representable number, then the resulting sequence is the sequence of Fibonacci numbers. 

This last point about the similarity with the Fibonacci numbers is interesting. Of course the Fibonacci numbers get very big very quickly while the Ulam numbers, because the smallest sum of any two previous numbers is sought, remain relatively smaller.

 Mathworld has a table of some of the other Ulam sequences:

I was struggling to get my SAGE code to generate these sequences of terms but came across some Python 2 code (source) that does the job, although I'm struggling to understand how it works:

ulams=[ ]
ulams.append(1)
ulams.append(2)
limit=100
sums=[0 for i in range(2*limit)]
newUlam=2
sumIndex=1
while(newUlam <limit):
  for i in ulams:
if (i<newUlam):
sums[i+newUlam] += 1
while(sums[sumIndex]!=1):
sumIndex += 1
newUlam = sumIndex
sumIndex += 1
ulams.append(newUlam) for i in ulams:
print i

With the limit set to 100, the following output is produced: 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

The code can be easily modified to generate other sequences with different starting terms e.g. 2 and 3:

ulams=[ ]
ulams.append(2)
ulams.append(3)
limit=100
sums=[0 for i in range(2*limit)]
newUlam=3
sumIndex=1
while(newUlam <limit):
  for i in ulams:
if (i<newUlam):
sums[i+newUlam] += 1
while(sums[sumIndex]!=1):
sumIndex += 1
newUlam = sumIndex
sumIndex += 1
ulams.append(newUlam) for i in ulams:
print i

This code produces the output: 2 3 5 7 8 9 13 14 18 19 24 25 29 30 35 36 40 41 46 51 56 63 68 72 73 78 79 83 84 89 94 115

ADDENDUM: I finally developed my own code which I do understand and it works! Here are the terms below 200 in the Ulam sequence beginning 2,3:

ulam=[i for i in range(1,200)]
ulam[0], ulam[1]=2,3
k=2
while k<len(ulam):
    count=0
    for x in range(k):
        for y in range(k):
            if ulam[x] > ulam[y]:
                if ulam[x] + ulam[y] == ulam[k]:
                    count+=1
    if count!=1:
        ulam.remove(ulam[k])
        k-=1
    k+=1
print ulam  

This code produces the output: 2, 3, 5, 7, 8, 9, 13, 14, 18, 19, 24, 25, 29, 30, 35, 36, 40, 41, 46, 51, 56, 63, 68, 72, 73, 78, 79, 83, 84, 89, 94, 115, 117, 126, 153, 160, 165, 169, 170, 175, 176, 181, 186, 191

Here is a link to an interesting article that looks at patterns in these 1-additive sequences.