A Ulam sequence with initial positive integers is the increasing sequence defined by and . For , is the least integer greater than having exactly one representation as the sum with . The numbers so produced are sometimes called u-numbers or Ulam numbers.

The first few numbers in the (1, 2)-Ulam sequence are 1, 2, 3, 4, 6, 8, 11, 13, 16, ... (OEIS A002858). Here, the first term after the initial (1, 2) is 3 since . The next term is . The representation is excluded because its summands are not distinct. 5 is not a member of the sequence since it is representable in two ways, , but is a member.

Proceeding in this manner, we can generate Ulam sequences for other initial ordered pairs , examples of which are given in the table below.

	Sloane	sequence
(1, 2)	A002858	1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
(1, 3)	A002859	1, 3, 4, 5, 6, 8, 10, 12, 17, 21, ...
(1, 4)	A003666	1, 4, 5, 6, 7, 8, 10, 16, 18, 19, ...
(1, 5)	A003667	1, 5, 6, 7, 8, 9, 10, 12, 20, 22, ...
(2, 3)	A001857	2, 3, 5, 7, 8, 9, 13, 14, 18, 19, ...
(2, 4)	A048951	2, 4, 6, 8, 12, 16, 22, 26, 32, 36, ...
(2, 5)	A007300	2, 5, 7, 9, 11, 12, 13, 15, 19, 23, ...

Schmerl and Spiegel (1994) proved that Ulam sequences for odd have exactly two even terms. Ulam sequences with only finitely many even terms eventually must have periodic successive differences (Finch 1991, 1992abc). Cassaigne and Finch (1995) proved that the Ulam sequences for (mod 4) have exactly three even terms.

The Ulam sequence can be generalized by the s-additive sequence.