Arithmetic Functions & Number Sequences

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Abstractions about number-theoretic transforms, sieves, series, densities, expansions, arithmetic functions, constants, and enumerative integer sequences.

16 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Binomial transform — An invertible triangular sequence transform that combines source terms with signed or unsigned binomial coefficients under a declared convention.
• Brun sieve — A combinatorial sieve that estimates integers avoiding specified prime divisibility conditions by truncating inclusion-exclusion with alternating upper and lower bounds.
• Catalan number — The integer sequence C_n=(1/(n+1)) binomial(2n,n) that counts many recursively nested structures such as balanced parentheses, polygon triangulations and binary trees.
• Dirichlet convolution — A binary operation on arithmetic functions defined by summing f(d)g(n/d) over the positive divisors d of n.
• Dirichlet density — An analytic density of a set of primes defined by its weighted prime Dirichlet series as the exponent approaches one from above.
• Dirichlet series — A complex series whose nth term is a coefficient multiplied by n raised to a complex negative exponent, serving as a multiplicative generating function in analytic number theory.
• Engel expansion — A representation of a positive real number as a sum of reciprocals of cumulative products from a unique nondecreasing integer sequence.
• Faulhaber's formula — A Bernoulli-number polynomial formula for the sum of equal nonnegative integer powers over the first n positive integers.
• Hurwitz zeta function — A two-parameter zeta function formed by summing inverse powers of an arithmetic progression and extended meromorphically beyond its defining half-plane.
• Irrationality measure — A quantitative bound on how closely an irrational real or complex number can be approximated by rational numbers as denominator size grows.
• Lambek–Moser theorem — A theorem constructing complementary integer sequences from generalized inverse nondecreasing functions.
• Lambert series — A generating series of the form sum a_n q^n divided by one minus q^n, whose expanded coefficients are divisor sums of the original sequence.
• Mertens function — The summatory function of the Möbius function, equal to the running difference between square-free integers with even and odd numbers of prime factors.
• Niven's constant — The limiting average, over positive integers, of the largest exponent in each integer’s prime factorization.
• Poly-Bernoulli number — A doubly indexed integer sequence defined by an exponential generating function involving the polylogarithm and generalizing Bernoulli numbers.
• Schuette–Nesbitt formula — A weighted generalization of inclusion–exclusion that expresses sums over outcomes with exactly or at least a given number of occurring events.