Divisor function — where it appears
Also named here as divisor sum, multiplicative function, perfect number — the same set of essays touches all of them, so they are one junction rather than several.
The shape of a number's divisors
Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.
Numbers that are their own parts
Six is one plus two plus three. Twenty-eight is one plus two plus four plus seven plus fourteen. Euclid explained where such numbers come from; Euler proved there are no others of that kind; and whether an odd one exists has been open for two thousand years.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysDivisor sumMultiplicative functionPerfect numberAbundanceExistence proofGeometric seriesHasse diagramLatticeMersenne primePrimesUnique factorisation