# Ulam number

The $n$th $U_{n}$ for $n>2$ is the smallest number greater than $U_{n-1}$ which is a sum of two smaller Ulam numbers in a unique way. $U_{1}=1$ and $U_{2}=2$; the sequence continues 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, etc. (listed in A002858 of Sloane’s OEIS); it is what is usually referred to as the Ulam sequence.

So, for example, 47 is an Ulam number because it is the sum of the pair of smaller Ulam numbers 11 and 36, and no other pair, while 48 is also an Ulam number because it is the sum of 1 and 47, and no other pair. 49 is not an Ulam number because it is the sum of 1 and 48, and of 2 and 47.

Stanisław Ulam (http://planetmath.org/StanislawUlam) first studied this sequences in the 1960s “in a peculiar attempt to get a 1D analog of a 2D cellular automaton  ” (Wolfram, 2002). In 2001 Jud McCranie verified that among the first 40000000, the only consecutive pairs that are also both Ulam numbers are 1 and 2, 2 and 3, 3 and 4, and 47 and 48. More recently, in 2006, Neil Sloane conjectured that a plot of the Ulam numbers will produce a line that is very close to

 $y=\frac{1351}{100}x.$

Ulam numbers and the resulting Ulam sequences can be generalized to having different initial values $U_{1}$ and $U_{2}$ with the only requirement being that $U_{1}, these are sometimes referred to as Ulam-type sequences. If $U_{1}=2$ and $2\nmid U_{2}$, then the Ulam-type sequence will have only one other even term (Schmerl & Spiegel, 1994).

## References

• 1 J. Schmerl & E. Spiegel, “The Regularity of Some 1-Additive Sequences”. J. Combinatoric Theory Ser. A 66 (1994): 172 - 175
• 2 S. Wolfram A New Kind of Science New York: Wolfram Media (2002): 908
Title Ulam number UlamNumber 2013-03-22 16:46:24 2013-03-22 16:46:24 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 11B13 Ulam sequence Ulam-type sequence