Ulam numbers

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 (the (1, 2)-Ulam sequence) starts with U₁ = 1 and U₂ = 2. Then for n > 2, U_n 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.

As a consequence of the definition, 3 is an Ulam number (1+2); and 4 is an Ulam number (1+3). (Here 2+2 is not a second representation of 4, because the previous terms must be distinct.) The integer 5 is not an Ulam number, because 5 = 1 + 4 = 2 + 3. The first few terms are

The first Ulam numbers that are also prime numbers 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: 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.

Ulam is said to have conjectured that the numbers have zero density, but they seem to have a density of approximately 0.07398.

It has been observed that the first 10 million Ulam numbers satisfy ${\displaystyle \cos {(2.5714474995a_{n})}<0}$ except for the 4 elements ${\displaystyle \left\{2,3,47,69\right\}}$ (this has now been verified up to ${\displaystyle n=10^{9}}$). 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 (see external links).

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