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Series

Perfect numbers — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · number
  2. 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.

    part 2 · number
  3. 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.

    part 3 · number
  4. 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.

    part 4 · number
  5. 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.

    part 5 · number

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