A plot of the Ulam numbers up to 108², arranged line by line in a square 108×108. The plot evidentiates the peaks in their density that occur with a frequency roughly equal to 108/5=21.6.

The members of the Ulam sequence are called Ulam numbers.

For example, after the first 4 terms which are trivially 1, 2, 3 and 4, the value of    cannot be 5, since    but it is 6, which can be obtained only as    (not as    because the terms added must be distinct).

The sequence is infinite because    is always a viable candidate for  .

Ulam numbers are not distributed uniformly, but their density has peaks at an average distance of 21.6 (see picture above).

The first Ulam numbers 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, 102 more terms

Below, the spiral pattern of Ulam numbers up to    . See the page on prime numbers for an explanation and links to similar pictures.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Ulam numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.