Number-Theoretic Sequences & Classes¶

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Abstractions about integers classified by divisibility, primality, representation, and arithmetic functions. They include special number families, perfectness and totients, modular computation, primality testing, multiplicative partitions, counting functions, and recursively generated sequences.

37 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.

Additive function — An arithmetic function satisfying f(ab)=f(a)+f(b) whenever positive integers a and b are coprime.
Arithmetic function — A function defined on positive integers, usually with complex values, that encodes a number-theoretic property of each integer.
Arithmetic number — A positive integer whose positive divisors have an integer arithmetic mean.
Carmichael number — A composite integer that satisfies the Fermat congruence for every integer base, so it systematically imitates prime behavior under the basic Fermat test.
Complete sequence — A sequence of natural numbers whose distinct finite subset sums represent every positive integer.
Composite number — A positive integer greater than one that can be expressed as a product of two smaller positive integers.
Cullen number — An integer of the form C_n = n·2^n + 1, forming a named exponential sequence whose rare prime terms are Cullen primes.
Erdős–Woods number — A positive integer k for which some interval of k+1 consecutive integers has every interior member sharing a nontrivial common divisor with at least one endpoint.
Euclid number — An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.
Euclid–Mullin sequence — A recursively defined prime sequence taking each new term as the least prime factor of one plus the product of all preceding terms.
Euler's totient function — The arithmetic function counting residue classes modulo a positive integer that are coprime to it.
Farey sequence — The increasing sequence of reduced rational numbers between zero and one whose denominators do not exceed a chosen order.
Friendly number — A positive integer sharing its abundancy index—the sum of divisors divided by the integer—with at least one distinct positive integer.
Gauss's lemma (number theory) — A criterion computing the Legendre symbol by counting how many least positive residues of a,2a,…,((p−1)/2)a modulo an odd prime exceed p/2.
Generalized taxicab number — The least integer expressible as a sum of a fixed number of positive k-th powers in a specified number of distinct ways.
Hemiperfect number — A positive integer whose sum-of-divisors function divided by the integer is a half-integer with odd numerator.
Highly composite number — A positive integer whose divisor count strictly exceeds that of every smaller positive integer.
Highly cototient number — A positive integer k>1 having more solutions to x−φ(x)=k than any smaller integer greater than one, where φ is Euler's totient function.
Highly powerful number — A powerful integer setting a new record for the product of its prime exponents among all smaller powerful integers.
Highly totient number — An integer whose number of preimages under Euler’s totient function exceeds that of every smaller integer.
Lucky number — A natural number surviving an iterative positional sieve that repeatedly deletes every kth remaining number.
Miller–Rabin primality test — A randomized strong-probable-prime test that repeatedly checks modular-power witnesses and bounds the chance that a composite integer passes all selected bases.
Modular exponentiation — Computation of a power modulo a positive integer, returning the residue of a base raised to an integer exponent without constructing the full power.
Multiplicative partition — An unordered factorization of a positive integer into integers greater than one, with products differing only by factor order identified.
Multiply perfect number — A positive integer whose sum of positive divisors is an integer multiple k of the number itself.
Noncototient — A positive integer that is not equal to n−φ(n) for any positive integer n, where φ is Euler's totient function.
Nonhypotenuse number — A natural number that is not the hypotenuse length of any integer-sided right triangle.
Odious number — A nonnegative integer whose binary expansion contains an odd number of one bits.
P-adic number — An element of the completion of the rational numbers under the non-Archimedean absolute value determined by a prime p.
Prime triplet — A set of three prime numbers spanning six integers, necessarily in one of two offset patterns apart from exceptional triples containing three.
Prime-counting function — The arithmetic function pi of x that counts prime numbers less than or equal to a real bound x.
Square number — An integer equal to the product of some integer with itself.
Størmer number — A positive integer whose squared value plus one has a prime factor at least twice the original integer.
Super-Poulet number — A composite base-two pseudoprime for which every positive divisor d also divides two-to-the-d minus two.
Supernatural number — A formal prime product whose exponent at each prime is a natural number or infinity, extending positive integers under divisibility.
Unusual number — A natural number whose largest prime factor is strictly greater than its square root.
Woodall number — A natural number of the form n times two to the n minus one.