Musing on a class of numbers

There is a little puzzle around in mathematical tuition sites that talks about polite numbers. These are numbers that are equal to the sum of a sequence of (at least two) consecutive positive numbers, so for example 2+3+4+5=14, so 14 is a polite number.

The question asks, "Which numbers are polite and which impolite?" I won't spoil it with an explicit answer.

But anyway, this got me thinking about another class of numbers, which I decided to call civil numbers. These are numbers which are "polite" by virtue of two consecutive sequences for example 15 = 4+5+6 = 7+8. Every formation sequence-sum is linked to an odd factor of the target number, either as the length of the sequence or as twice the centre-value: so 4,5,6 links to 3 and 7,8 links to 15. Also each odd factor generates exactly one sequence sum. As I explored these civil numbers it became clear that I needed to allow a sequence of length 1 also, which extended the class of civil numbers by exactly one member: 3 = 1+2 = 3.

Some fun facts:

• Only 15 and 3 use a sequence of length 2 in their formation sum.


• Any civil number can be multiplied by an odd square to produce another civil number.


• Even disqualifying such "derived" civil numbers, there are an infinite number of civil numbers including...


• for each consecutive pair of numbers, there is a civil number (one only) composed of adjacent sequences of those lengths. Example, at random: 315 and 316 length sequences are used to form 31404870 = (99225+...+99540) = (99541+...+99855)
  This is also the highest civil number that uses a 315-length sequence.


• All civil numbers are divisible by 3 (proven, if a little tricky)


• Doubly and n-fold civil numbers exist (105 is the smallest doubly-civil) composed of more than one pair of adjacent sequences, and can also be multiplied by any odd square to produce another such.

I also looked for the elusive gallant numbers (which may well not exist), which are "polite" by virtue of three consecutive sequences. 42 (which is civil) is a near-miss for gallantry, I guess, at 42 = 3+4+5+6+7+8+9 = 9+10+11+12 = 13+14+15 , with the first two sequences overlapping at 9. I should warn anyone contemplating renewing my search that I have eliminated the possibility of such a number exisiting that is less than 100 million. If any exist, then of course there are an infinite number of these too, because again multiplying by any odd square would produce another one.

I see Dr Ron Knott has some interesting pages on what he calls Runsums. Civil numbers are called "Neighbourly Runsums".

Vehicle hostility

Ok, driving is the most dangerous activity we do. I understand that, do what I can to reduce that risk to me.

The market has a perfectly viable lever, which it uses pretty effectively these days, to sell us all cars which will keep the occupants of the cars safer . Crumple zones, airbags, auto unlocking, all wonderful stuff.

However. There is no marketing edge, as yet anyway, in producing cars which make other road users safer, or at least less threatened. I fear that pressure towards cars that are less damaging towards other road users must initiate with some form of legislation. After all, at present there is no substantial disadvantage to owning and using a car which is going to inflict maximal damage on any other cars or other road users unfortunate to be involved in a collision with it.

Therefore I propose a measure of the damage which a vehicle inflicts on other vehicles. This measure would probably be known formally be some mouthful of a name like "third-party repercussion appraisal rating" but for the purpose of this discussion I'll call "vehicle hostility". This rating could then be used (for example) as a multiplier on any fines arising from antisocial road use - speeding, running red lights, failure to indicate, etc.

The components of the rating should eventually be statistically demonstrated damage enhancers, but that might take too long given the present state of accident information. In my scheme, hostility starts at a value of 1, standard acceptable hostility. I'm not sure if vehicles that are less hostile than "standard" should gain anything but I can see an argument for allowing trade-off between different factors, eg. high weight and low rigidity. I'll propose some components that feel like they ought to play a part:

Weight: if weight >1000kg, multiply hostility by weight/10000kg
Width: Similar for over-standard width
Overhang: If the vehicle extends outside wheel base by a significant amount this should also increase hostility
Rigidity: I'm not sure how to measure this but it also plays a part.
Clearance: Vehicles high off the road will do more damage
Fittings: Standard hostility multipliers for bull (roo) bars and other injury-enhancing fittings
Articulation: Towed or flexing will generally increase the chance of injuring/damaging more than otherwise.

With time this might allow the fear of increased penalties to drive vehicles in the direction of doing less damage to others as well as protecting the occupants.

What is this thing called Blog?

I thought I'd try putting up a few posts, anyway. I should post at least every two weeks. I may even develop some specific themes.

At the moment I spend a lot of time prowling around nuclear power supporting sites, and I must make an effort to do more in real life... and starting this probably won't help either.

I should post a short health warning for the Questionable Content comic link - I visited this brilliant creation of Jeph Jacques' about a year ago and spent about 20 hours straight through the night reading the archive. If it grabs you, it's addictive. The hardest part is finishing the archive - then you have to wait a whole day (or the whole weekend) for the next installment!