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An Ulam number is a member of an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964.^[1] 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.

Contents [hide] 1 Examples 2 Infinite sequence 3 Generalizations 4 Notes 5 References 6 External links

[edit]Examples

By 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 (sequence A002858 in OEIS).

The first Ulam numbers that are also prime numbers are

2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489 (sequence A068820 in OEIS).

[edit]Infinite sequence

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