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Concept

Divisor sum

The total of all a number's divisors, the number itself included. Comparing it with twice the number itself is what sorts numbers into deficient, perfect and abundant, and Euclid's rule builds a perfect one from every Mersenne prime.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

Also named here as perfect number — the same set of essays touches all of them, so they are one junction rather than several.

The divisors of 60. Every divisor as a lattice point, one axis per prime, joined when one divides the other by a single prime.

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.

number · Unique factorisation 28 is perfect, because its divisors form this rectangle. Two rows of divisors: the powers of two, and the same powers multiplied by the Mersenne prime.

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.

number · Perfect numbers Where the aliquot sequence of every number up to 1000 goes. A grid of the starting values 2 to 1000 coloured by the fate of each aliquot sequence: 964 reach 1, 19 reach a perfect number, 3 enter a cycle, and 13 pass 10²² undecided.

The sum of the parts, taken again

Replace a number by the sum of its proper divisors and do it again. Most numbers fall to 1, a few land on a perfect number or a cycle, and some climb for hundreds of steps. Below a thousand there are twelve whose fate nobody knows — and the thing that keeps them climbing is a perfect number hiding in their factorisation.

number · Perfect numbers How the abundancy σ(n)/n is distributed, up to 1,000,000. The share of numbers up to 1000000 whose ratio σ(n)/n is at least t, for t from 1 to 4. It is 0.2475 at two, 0.0202 at three and 0.00023 at four, with a steep fall just above one, where the numbers with no small prime factor sit.

Why a quarter of numbers overshoot

About one number in four has proper divisors adding to more than itself. That a proportion exists at all is not automatic — there are sets defined just as simply that have no proportion — and the reason this one does is that abundance is inherited by multiples, and the numbers it is first inherited from are sparse enough to add up.

number · Perfect numbers Numbers whose divisors add to two, three, four, five and six times themselves. A table of multiperfect numbers with the multiple their divisor sum makes of them, the number itself and its factorisation into prime powers.

Divisors that add to three times the number

The divisors of 6 add up to 12, twice 6: a perfect number. The divisors of 120 add up to 360, three times 120, and those of 30,240 to four times it. Numbers like these were a sport for Fermat and Descartes, and they are held together by one fact — the ratio σ(n)/n is a product over the primes, and each prime can add only a little.

number · Perfect numbers Numbers up to 600 that share their abundancy. A scatter of abundancy against n with horizontal segments joining numbers that have exactly the same abundancy.

A ratio nobody else has

Divide the sum of a number's divisors by the number and you get its abundancy: 2 for every perfect number, 12/5 for both 30 and 140. Numbers that share an abundancy are called friends. Some numbers provably have no friend at all, most have friends only far away — and for 10, whose abundancy is 9/5, nobody knows whether a friend exists.

number · Perfect numbers Descartes' number, perfect but for one factor. Five boxes for the prime powers of Descartes' number, each with its divisor sum factored beneath, multiplying to exactly twice the number provided 22021 is treated as a prime.

Perfect but for one factor

In 1638 Descartes wrote to Mersenne with an odd number whose divisors add up to exactly twice the number — provided one of its factors, 22021, is counted as a prime. It is not; it is 19² × 61. Nearly four centuries later that number is still the only odd one of its kind known, and a search of every odd number below ten million finds no other. It is the closest anyone has come to an odd perfect number, and what it shows is how an odd perfect number would have to be built.

number · Perfect numbers Seventy, and the totals its divisors can make. A strip of the totals 0 to 74 marking those reachable as sums of distinct proper divisors of 70; 70 itself and the excess 4 are not.

Abundant, and still not a sum of its parts

Seventy's proper divisors add to seventy-four, more than seventy itself — and yet no selection of them adds to exactly seventy. The reason is that the excess, four, cannot be made from the small divisors one, two and five. Numbers like seventy are called weird; there are thirty-six below twenty thousand, every one of them even, and nobody knows whether an odd one exists.

number · Perfect numbers Twelve's divisors make every total up to twenty-eight. Stacked bars for each total from 1 to 28, each built from distinct divisors of 12.

Divisors that make every amount

Twelve's divisors — 1, 2, 3, 4, 6, 12 — can be chosen to add to every whole number from one to twenty-eight. Numbers like that are called practical, a test on their primes decides them, and they turn out to be counted like the primes: about 1.336 x / log x of them below x. For practical numbers Goldbach's conjecture and the twin conjecture are theorems.

number · Perfect numbers

Named alongside it

The objects these essays reach for when they reach for this one.

Perfect numberAbundanceMultiplicative functionExhaustive searchPrimesDivisor functionDensityCounterexampleCounting two waysMersenne primeOpen problemParity

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