Integer Classifications & Arithmetic Functions

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Abstractions about classifying integers by divisor-sum and multiplicative properties (perfect, deficient, hyperperfect, sublime numbers), named integer sequences (Fermat, Cullen, Thabit numbers), arithmetic functions (totient, Möbius, Dedekind psi), and related notions like modular inverses and factorization.

43 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.
• Bhargava factorial — A factorial sequence attached to an arbitrary subset of the integers through p-orderings, generalizing n-factorial while preserving divisibility and integer-valued-polynomial properties.
• Complex-base system — Represent real or complex numbers positionally using a nonreal radix and a finite digit alphabet, allowing a single unsigned expansion to encode multiple dimensions when admissibility, convergence, and uniqueness conditions hold.
• 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.
• Cunningham number — An integer of the form b^n−1 or b^n+1 with integer base b that is not itself a perfect power, organizing prominent exponential families for factorization and primality study.
• Cyclic number (group theory) — A positive integer n such that every group of order n is cyclic, equivalently n is coprime to Euler's totient phi(n).
• Dedekind psi function — The multiplicative arithmetic function ψ(n)=n times the product of (1+1/p) over the distinct prime divisors of n.
• Deficient number — Classify a positive integer as deficient when the sum of its positive proper divisors is smaller than the integer itself, equivalently when its divisor sum is less than twice the integer.
• Diophantine quintuple — A five-element set of positive integers for which the product of every two distinct elements plus one is a perfect square.
• Dirichlet convolution — A binary operation on arithmetic functions defined by summing f(d)g(n/d) over the positive divisors d of n.
• Division (mathematics) — Recover a quotient q from dividend a and nonzero divisor b by solving bq=a, with exact, remainder, rational, field and algorithmic meanings determined by the ambient number system.
• Engel expansion — A representation of a positive real number as a sum of reciprocals of cumulative products from a unique nondecreasing integer sequence.
• Euclid number — An integer one greater than the product of the first n primes, with a second-kind variant one less than that primorial.
• Euler's totient function — The arithmetic function counting residue classes modulo a positive integer that are coprime to it.
• Fermat number — Generate the integer sequence F_n = 2⁽²n) + 1, whose product recurrence makes distinct terms pairwise coprime and whose rare prime members connect to constructible polygons.
• Fortunate number — For each positive index n, select the least integer m greater than one for which the nth primorial plus m is prime, producing the sequence governed by Fortune's still-open primality conjecture.
• 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.
• Hyperperfect number — Classify a natural number by the parameterized divisor-sum constraint n = 1 + k(σ(n) − n − 1), with perfect numbers as the k = 1 boundary case.
• Integer factorization — The decomposition of a positive integer into integer factors, canonically into a unique multiset of primes up to ordering, with computational difficulty depending strongly on input size and structure.
• Irrationality measure — A quantitative bound on how closely an irrational real or complex number can be approximated by rational numbers as denominator size grows.
• Kaprekar number — Classify a base-b natural number whose square can be split at a declared digit position into two parts that sum back to the number.
• Knödel number — For a fixed positive integer n, a composite integer m such that a^(m−n) is congruent to one modulo m for every integer a coprime to m.
• Lunar arithmetic — Replace digit addition by maximum and digit multiplication by minimum, extending them positionally without carries so nonnegative base-b numerals form a closed idempotent arithmetic with altered sums, products, factors, and primes.
• Modular multiplicative inverse — A residue x whose product with a given integer a is congruent to one modulo m, existing exactly when a and m are coprime.
• Muckenhoupt Weights — Positive Euclidean weights whose local mean and reciprocal-power mean satisfy a uniform, exponent-indexed A_p balance over all cubes.
• Multiplicative Function — Recognize an arithmetic function normalized at one whose value on a product of coprime positive integers factors into the product of their values, thereby reducing global behavior to independently specified prime-power data.
• Möbius function — The multiplicative arithmetic function μ(n) that is zero on numbers divisible by a prime square and otherwise equals minus one to the number of distinct prime factors.
• Noncototient — A positive integer that is not equal to n−φ(n) for any positive integer n, where φ is Euler's totient function.
• Partition function (number theory) — The arithmetic function p(n) that counts unordered representations of a nonnegative integer as a sum of positive integers, with generating-function, recurrence, asymptotic and modular-congruence structure.
• Perfect number — Classify a positive integer as perfect when the sum of all its proper positive divisors equals the integer itself, equivalently when its divisor-sum is exactly twice the integer.
• Prime signature — Classify a positive integer by the unordered multiset of positive exponents in its unique prime factorization, discarding prime labels while preserving multiplicative shape.
• Quadratic residuosity problem — The computational decision problem of determining whether a number with Jacobi symbol one is a square modulo a composite whose prime factorization is unknown.
• Quasiperfect number — Classify a positive integer by the divisor-sum equation sigma of n equals twice n plus one, an unresolved class whose hypothetical members must satisfy strong odd-square and prime-factor restrictions.
• Ramanujan's sum — The finite exponential sum c_q(n) over residues coprime to q, an integer-valued arithmetic function used as a Fourier basis for number-theoretic expansions.
• Reciprocity law — A number-theoretic rule relating splitting behavior of primes in an algebraic extension to congruence or residue information, generalizing quadratic reciprocity.
• Reduced residue system — A complete set of incongruent representatives modulo n chosen from the integer classes coprime to n.
• Sublime number — A positive integer having a perfect number of positive divisors and a divisor sum that is itself a perfect number.
• Subtraction — An arithmetic operation that obtains the difference between a minuend and subtrahend, ordinarily defined as addition of the subtrahend's additive inverse where that inverse exists.
• Sum of squares function — Count ordered signed integer k-tuples whose squared coordinates sum to n, yielding the arithmetic function r_k(n) and its divisor-sum, theta-series, and local-obstruction structure.
• Thabit number — An integer of the form 3·2^n−1 for a nonnegative integer n, historically linked through special primality conditions to constructions of amicable numbers.
• Tunnell's theorem — Test a square-free integer for the congruent-number property through equalities among counts of representations by four ternary quadratic forms—necessary unconditionally and sufficient conditional on Birch–Swinnerton-Dyer.
• Wall–Sun–Sun prime — A conjectural prime p for which p² divides the Fibonacci number indexed by p's Pisano period, equivalently a Fibonacci–Wieferich prime under standard formulations.
• Wilson quotient — For a prime p, the integer ((p−1)!+1)/p, whose residues encode refinements of Wilson's theorem and define Wilson primes when divisible by p.