In mathematics, the Ulam numbers comprise 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, 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 (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 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, ... . 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. Proof: Assume that for n > 2, Un−1 + Un = Un+1 is the required sum in only one way; then so does Un−2 + Un produce a sum in only one way, and it falls between Un and Un+1. This contradicts the condition that Un+1 is the next smallest Ulam number. For n > 2, any three consecutive Ulam numbers (Un−1, Un, Un+1) as integer sides will form a triangle. Proof: The previous property states that for n > 2, Un−2 + Un ≥ Un + 1. Consequently Un−1