Ulam number

The $n$th Ulam number $U_n$ 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 , 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).

Title	Ulam number
Canonical name	UlamNumber
Date of creation	2013-03-22 16:46:24
Last modified on	2013-03-22 16:46:24
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	6
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11B13
Defines	Ulam sequence
Defines	Ulam-type sequence