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Abbreviating the number of constants as f 0, and the number of i-ary function symbols as f i, the number θ h of distinct ground terms of a height up to h can be computed by the following recursion formula: θ 0 = f 0, since a ground term of height 0 can only be a constant, + = =, since a ground term of height up to h+1 can be obtained by ...
The two formulas were described by the Irish monk Dicuil in about 816 in his Computus. [5] An English translation of Dicuil's account is available. [6] Occasionally it is necessary to compute large triangular numbers where the standard formula t = n*(n+1)/2 would suffer integer overflow before the final division by 2.
Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
The prime number race generalizes to other moduli and is the subject of much research; Pál Turán asked whether it is always the case that π c,a (x) and π c,b (x) change places when a and b are coprime to c. [34] Granville and Martin give a thorough exposition and survey. [33] Graph of the number of primes ending in 1, 3, 7, and 9 up to n ...
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. Formulas for calculating primes do exist; however ...
Ramanujan summation is a method to isolate the constant term in the Euler–Maclaurin formula for the partial sums of a series. For a function f , the classical Ramanujan sum of the series ∑ k = 1 ∞ f ( k ) {\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)} is defined as
The nth pentagonal number p n is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of ...
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. [ 1 ] [ 2 ] For example, in a signature consisting of a single binary operation , the term algebra over a set X of variables is exactly the free magma generated by X .