[seqfan] Re: Numbers of the form n*(n+k)

Marc LeBrun mlb at well.com
Mon Mar 23 21:41:32 CET 2020

> MARION <charliemath at optonline.net> 
> to refer to "numbers of the form n(n+k), I will be using "k-oblongs."

OK, nice, let's say "n is k-oblong" when there's a k >=0 with n = m (m+k).  

> Neil Sloane <njasloane at gmail.com> 
> There is an old law that "thou shalt not create new terminology unnecessarily".

Heh, now I can't resist!

Define the least such k, call it c, the "coolness" of n: A056737.
And while we're lawbreaking, let's call m the "breadth" and m+c the "length" of n: A033676 and A033677.

Dig it: squaresville is the least cool, but primes are the mostest!

Can we give an asymptotic formula for the coolness?
Similarly, what is the average behavior of the "aspect ratio" breadth/length = m/(m+c)?
Or the semiperimeter-to-area: (2m+c)/n?

Let's use a pipe "|" to show the coolest division of n: breadth | length.

If we denote the non-decreasing prime factors of n with PQRST... (eg 90 = 2,3,3,5)
then the primes can obviously be expressed as 1 | P, and "biprimes" are always P | Q.
The triprimes A014612 are always either PQ | R or the swap R | PQ (eg 42 and 30 resp.)
When we get to four primes it's always either PQR | S or QR | PS (or their swaps),
For a given number of primes, how many patterns are there?

For more fun we can generalize oblong coolness to higher-dimensional integral bricks by saying the coolest division gives the most compact brick, with the least surface-to-volume ratio, or alternatively, minimal bounding hypercube, or equivalently simply the smallest maximum side-length, etc.  What are the asymptotics and possible prime pipe patterns there?

Also, could it make sense to generalize to factorizations over quadratic extensions?  Minimizing coolness may not work, but might conjugate factors cancel their non-integral parts when summed into a semiperimeter analog?

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