Integer Functions & Special Numbers

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Abstractions about multiplicative functions, factorization, partitions, prime signatures, and named classes of integers defined by divisor properties.

13 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.

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• Sublime number — A positive integer having a perfect number of positive divisors and a divisor sum that is itself a perfect number.