Ulam numbers

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A Ulam number is a member of an integer sequence which was devised by Stanislaw Ulam and published in SIAM Review in 1964. The standard Ulam sequence starts with U₁=1 and U₂=2 being the first two Ulam numbers. 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 (Guy 2004:166-67). Ulam conjectured that the numbers have zero density, but they seem to have a density of approximately 0.07396.

1 Examples
2 Generalization
3 References
4 External links

[edit] Examples

By the definition, 3=1+2 is a Ulam number; and 4=1+3 is a Ulam number (The sum 4=2+2 doesn't count because the previous terms must be distinct.) The integer 5 is not a 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, and 99. (sequence A002858 in OEIS).

[edit] Generalization

The idea can be generalized by selecting different starting values and by requiring that the terms be a sum of previous terms in a given number of ways.