Integer Properties & Number-Theoretic Sets

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Abstractions about special classes and properties of integers, covering divisor and factorization conditions (smooth numbers, square-free elements, refactorable and untouchable numbers), additive and density-based sets (sum-free and primefree sequences, Schnirelmann density), and named integer classifications like Giuga and Descartes numbers.

14 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.

• Coin Problem — Find the largest nonnegative integer not representable as a nonnegative integer combination of given coprime positive denominations—the Frobenius number of their numerical semigroup.
• Descartes Number — An odd composite integer equipped with a coprime factorization D=km that satisfies the perfect-number divisor-sum equation only when the composite factor m is formally treated as a single prime-like factor.
• Fundamental Theorem of Arithmetic — The integer-specific theorem that every integer greater than one is a product of primes and that its prime multiset is unique up to order.
• Giuga Number — A composite integer n for which every prime divisor p satisfies p dividing n/p minus 1, a restrictive factorwise congruence linked to Giuga's primality conjecture.
• Pillai's Arithmetical Function — The multiplicative gcd-sum function P(n)=sum from k=1 to n of gcd(k,n), whose divisor-class decomposition P=id*phi exposes prime-power evaluation, Euler products, and average-order analysis.
• Primefree Sequence — A nontrivial Fibonacci-type integer sequence begun from coprime composite seeds and proved to contain only composite terms, typically by a finite cover of periodic modular divisibility classes.
• Refactorable number — Classify a positive integer as refactorable when the number of its positive divisors divides the integer itself.
• Schnirelmann Density — Measure a set of positive integers by the least fraction it occupies in any initial segment, making early gaps permanently visible and enabling quantitative sumset and additive-basis theorems.
• Semiperfect Number — A positive integer that equals the sum of a distinct subset of its proper divisors, whether or not the subset uses them all.
• Smooth Number — A positive integer is B-smooth when every prime divisor is at most the declared bound B, making its factorization lie entirely within a small-prime factor base.
• Sparsely Totient Number — A natural number n whose Euler totient is a strict suffix minimum: every larger integer m has φ(m) greater than φ(n).
• Square-Free Element — A nonzero element of a unique factorization domain whose irreducible factors all have multiplicity at most one, equivalently one not divisible by the square of any nonunit.
• Sum-Free Sequence — A strictly increasing sequence of positive integers in which no term is representable as a sum of a subset of its predecessors, coupling prefix-dependent additive avoidance to sparse-growth and reciprocal-sum questions.
• Untouchable Number — A positive integer lying outside the image of the aliquot-sum function, classified by the nonexistence of any positive integer whose proper divisors sum to it.