Ulam sequences

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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. 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.

A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n > 2, a(n) = least number > a(n - 1) which is a unique sum of two distinct earlier terms.

{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, ...}

A068820 Ulam numbers that are primes.

{2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897, 3041, 3109, 3217, 3371, ...}

Examples

By definition of the (1, 2)–Ulam sequence, 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.

References

• Ulam, Stanislaw (1964a), “Combinatorial analysis in infinite sets and some physical theories”, SIAM Review: 343–355 .
• Ulam, Stanislaw (1964b), Problems in Modern Mathematics, Wiley-Interscience, p. xi .