A005187

A005187 is an integer sequence in the On-Line Encyclopedia of Integer Sequences (OEIS). The sequence is defined by the property that its terms are the number of partitions of n into distinct parts. A partition of an integer n into distinct parts is a way of writing n as a sum of positive integers where each integer in the sum is unique. For example, the partitions of 5 into distinct parts are 5, 4+1, and 3+2. Therefore, the number of partitions of 5 into distinct parts is 3. The first few terms of the sequence are 1, 1, 2, 2, 3, 4, 5, 6, 8, 10.

This sequence is also known as the partition function q(n). It has connections to other areas of mathematics, including combinatorics and number theory. The generating function for this sequence is given by the infinite product $\prod_{k=1}^\infty (1+x^k)$. This means that when the infinite product is expanded as a power series, the coefficient of $x^n$ is the n-th term of the sequence A005187. The study of partition functions and their properties has a rich history, with significant contributions from mathematicians like Euler and Ramanujan. The sequence A005187 provides a specific count for a particular type of integer partition, highlighting the combinatorial diversity found in number theory.