#superabundant

Oregon Public Broadcasting has an entire series of minidocs about local ingredient, where they come from, and all the ways that food tells the story of Oregon history.

Extremely cool to be sitting on the couch eating Eugene-made Pasta Plus large shells, from Eugene-made Camas Mills four, made from locally grown durum wheat, while public television talks about why the wheat is special (and interviews the Camas Mills owners).

I have never in my life heard the narrator of an educational program sound so hungry to try eating our way out of an ecological imbalance problem (and disappointed that we probably can't). A sea otter wrote this.

What unites these bodies of work is the density of drawing across the picture surface, presenting numerous stories to be unpicked or decoded. Looking at these works is similar to reading a page of a newspaper. Each part of the page contains different story, report or account – set out in tightly printed text – though these individual narratives collectively build a complex picture of the world on a particular day. Likewise the eye moves across the surface of Jan and Phil’s drawings stopping here and there to discover a strange or beautiful moment connected to the whole and yet somehow equally independent of the busyness of the pictorial world constructed.

William Blake himself, who had his first vision at Peckham Rye wrote: ‘To see a World in a Grain of Sand And a Heaven in a Wild Flower Hold Infinity in the palm of your hand And Eternity in an hour’

A002093 - Highly abundant numbers

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, ...

A004394 - Superabundant numbers

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ...

The sum of an integer's proper divisors is represented as σ(n). [Please see the correction below.] If σ(n) > σ(m) is true for for all m < n, that number is called highly abundant. If σ(n)/n > σ(m)/m is true for for all m < n, that number is called superabundant. It's almost trivially apparent that all superabundant numbers are also highly abundant, and that there are numbers which are highly abundant but not superabundant (for instance: 10 and 16). Leonidas Alaoglu and Paul Erdős's early work on these numbers, however posed a question: Is there an infinite number of integers that are highly abundant but not superabundant? (Spoiler alert: There are. Jean-Luc Nicolas proved it in 1969).

Interestingly, not all of these numbers are even necessarily abundant. Since σ(2) = 3 and σ(4) = 7, they both meet the criteria for both of these sequences. But σ(2) - 2 = 1 and σ(4) - 4 = 3, making them both deficient. [This paragraph was originally both wrong and confusing. Please see the correction below.]

There's a complex relationship between superabundant numbers and the Riemann hypothesis, which says a lot about the distribution of prime numbers (not to mention other, deeper fields of number theory).

Correction: Thanks to orthogonality on MetaFilter, I've become aware that my definition of the divisor function σ(n) was wrong. It is the sum of all divisors, including n. The restricted divisor function, which includes only the proper divisors (those that are not n) is denoted by s(n). My apologies for the mistake.