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The values (), …, of the partition function (1, 2, 3, 5, 7, 11, 15, and 22) can be determined by counting the Young diagrams for the partitions of the numbers from 1 to 8. In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the ...
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
2 + 2 + 1; 2 + 1 + 1 + 1; 1 + 1 + 1 + 1 + 1; Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple (2, 2, 1) or in the even more compact form (2 2, 1) where the superscript indicates the number of repetitions of a ...
In plain words, e.g., the first congruence means that If a number is 4 more than a multiple of 5, i.e. it is in the sequence 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later other congruences of this type were discovered, for numbers and for Tau-functions.
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy [1] states that the normal order of the number () of distinct prime factors of a number is . Roughly speaking, this means that most numbers have about this number of distinct prime factors.
The Hardy–Ramanujan theorem: the normal order of ω(n), the number of distinct prime factors of n, is log(log(n)); The normal order of Ω(n), the number of prime factors of n counted with multiplicity, is log(log(n)); The normal order of log(d(n)), where d(n) is the number of divisors of n, is log(2) log(log(n)).
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. [2] [3]
By a partition of a positive integer n we mean a finite multiset λ = { λ k, λ k − 1, . . . , λ 1} of positive integers satisfying the following two conditions: . λ k ≥ . . . ≥ λ 2 ≥ λ 1 > 0.