An Ulam number is a member of an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence starts with U1 = 1 and U2 = 2. Then for n > 2, Un is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way and larger than all earlier terms.
Examples
As a consequence of the definition, 3 is an Ulam number ; and 4 is an Ulam number. The integer 5 is not an Ulam number, because 5 = 1 + 4 = 2 + 3. The first few terms are 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: 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. Ulam is said to have conjectured that the numbers have zero density, but they seem to have a density of approximately 0.07398.
Properties
Apart from 1 + 2 = 3 any subsequent Ulam number cannot be the sum of its two prior consecutive Ulam numbers. For n > 2, any three consecutive Ulam numbers as integer sides will form a triangle. The sequence of Ulam numbers forms a complete sequence. For every integer n > 1 there is always at least one Ulam number Uj such that n ≤ Uj < 2n. In any sequence of 5 consecutive positive integers there can be a maximum of 2 Ulam numbers.
Inequalities
Ulam numbers are pseudo-random and too irregular to have tight bounds. Nevertheless from the properties above, namely, at worst the next Ulam number Un+1 ≤ Un + Un-2 and in any five consecutive positive integers at most two can be Ulam numbers, it can be stated that where Nn are the numbers in Narayana’s cows sequence: 1,1,1,2,3,4,6,9,13,19,... with the recurrence relationNn = Nn-1 +Nn-3 that starts at N0.
Hidden structure
It has been observed that the first 10 million Ulam numbers satisfy except forthe four elements . Inequalities of this type are usually true for sequences exhibiting some form of periodicity but the Ulam sequence does not seem to be periodic and the phenomenon is not understood. It can be exploited to do a fast computation of the Ulam sequence.
Generalizations
The idea can be generalized as -Ulam numbers by selecting different starting values. A sequence of -Ulam numbers is regular if the sequence of differences between consecutive numbers in the sequence is eventually periodic. When v is an odd numbergreater than three, the -Ulam numbers are regular. When v is congruent to 1 and at least five, the -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 -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.
