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The nth partial sum is given by a simple formula: = = (+). This equation was known to the Pythagoreans as early as the sixth century BCE. [5] Numbers of this form are called triangular numbers, because they can be arranged as an equilateral triangle.
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
The concept of a decimal digit sum is closely related to, but not the same as, the digital root, which is the result of repeatedly applying the digit sum operation until the remaining value is only a single digit. The decimal digital root of any non-zero integer will be a number in the range 1 to 9, whereas the digit sum can take any value.
a perfect number equals the sum of its proper divisors; that is, s(n) = n; an abundant number is lesser than the sum of its proper divisors; that is, s(n) > n; a highly abundant number has a sum of positive divisors that is greater than any lesser number; that is, σ(n) > σ(m) for every positive integer m < n. Counterintuitively, the first ...
Thus, there are a finite number of sum-product numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point. The number of iterations i {\displaystyle i} needed for F b i ( n ) {\displaystyle F_{b}^{i}(n)} to reach a fixed point is the sum-product function's persistence of n ...
The abundancy index of n is the ratio σ(n)/n. [7] Distinct numbers n 1, n 2, ... (whether abundant or not) with the same abundancy index are called friendly numbers. The sequence (a k) of least numbers n such that σ(n) > kn, in which a 2 = 12 corresponds to the first abundant number, grows very quickly (sequence A134716 in the OEIS).
These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.
This is a list of all articles about natural numbers from 1 to 10,000. Red links are included to make it clearly visible which articles exist and which do not. Existing articles can either be articles with content, or redirects .