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Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula = (,) =,where (a, q) = 1 means that a only takes on values coprime to q.
The nth partial sum of the series is the triangular number ∑ k = 1 n k = n ( n + 1 ) 2 , {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},} which increases without bound as n goes to infinity .
and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3) 2 / 12. [ 14 ] Partitions in a rectangle and Gaussian binomial coefficients
In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. [1] The most famous taxicab number is 1729 = Ta(2) = 1 3 + 12 3 = 9 3 + 10 3 , also known as the Hardy-Ramanujan number.
The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts. The number q(n) is also equal to the number of partitions of n in which only odd summands are permitted. [20]
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number, expressed as the sum of two cubic numbers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan.
Thus () gives the number of decays of an integer n in which adjacent parts of the partition differ by at least 2 and in which the smallest part is greater than or equal to 2 is equal the number of decays whose parts are equal to 2 or 3 mod 5. This will be illustrated as examples in the following two tables: