
August 11th, 2013
12:13 pm  Some sequences of numbers that I really like OK, this is geeky even by my standards, but you expect nothing less, right? It's too geeky for my Facebook and Twitter audiences, but I feel safe here.
10, 16, 25, 40, 64, 100, 160, 250, 400, 640, 1 000, 1 600, 2 500, ... (each later item is ten times the one five previous)
The powers of 2 and 5 in this sequence give it a particularly attractive feel to me, and I am deliberately using deliberately fluffy, imprecise language here that I do not feel the need to justify this on any more than the vaguest of aesthetic grounds.
It is not a geometric progression, with the ratios between successive powers being 2^{3}/5, 5^{2}/2^{4}, 2^{3}/5, 2^{3}/5, 5^{2}/2^{4}, (etc.) but 25/16 = 1.5625 is handwavingly not too far from 1.6 to give it a reasonably consistent sort of feel. Additionally, the ratio between powers and their nextbutone values are 2.5, 2.5, 2.56, 2.5 and 2.5, etc., and the ratio between powers and their nextbut two values are 4, 4, 4, 4 and 3.90625 etc. OK, these aren't all the same as would be the case in a geometric progression, but they're mostly really close, and isn't that cool?
The sequence was, putting it politely, adapted from Herman's Top Olympians scoring system, and bears a distinct resemblance to the R5 sequence of Renard numbers  in fact, it essentially is the R5 sequence of Renard numbers except that there they replace 64 with 63. Now I reluctantly conclude that 10^{0.8} is closer to 6.3 than it is to 6.4, making the ratios between successive members of R5 closer to each other than the ratios in my sequence above and with similar knockon effects to the other properties. However, I choose to prefer increased frequency of repetition at the cost of making the absolute difference between some ratios a little higher. Similarly, giving preference to the fact that all the numbers have only prime factors of 2 and 5 is arbitrary, and there may well be situations where the inclusion of 63 = 3*3*7 is a useful factorisation. I choose not to care.
The whole phenomenon of preferred numbers is quite fun. I enjoyed spotting the similarity between the R10" progression and the progression of the blind structure in the World Series of Poker main event from about level five onwards. I don't know if this was independent reinvention (or, perhaps more likely, redevelopment) by coincidence or deliberate, but it goes to show a practical use of the principle. I don't claim there particularly needs to be a practical use for any of this, but this might be one, and issues of coinage selection in currency design also present themselves here as well.
I also have long had a liking for the sequence
1, 2, 4, 10, 30, 100, 400, 2 000, 12 000, 100 000, 1 000 000 (and not yet really defined after that)
because it again has a strong focus on factors of 2 and 5, with only a couple of incidental 3s, but also has a somewhat factoriallike nature whilst there is the additional property that I'm really attracted to whereby only one of the sequence members has more than a single significant figure  and "12" about as friendly and familiar as twosignificantdigit numbers get. Again, no particular reason for this other than a vague claim to aesthetic neatness, but you just might agree with me that it nonspecifically feels quite neat.
The ratios between subsequent members are 2, 2, 5/2, 3, 10/3, 4, 5, 6, 25/3 and 10. Other than the first, these are strictly increasing, and the size of the increase is not very far off being strictly increasing itself. This increase is slower at first than the factorial sequence (which is, by definition, 1, 2, 3, 4, ...) but does speed up towards the end.
Again, I don't claim to have a practical use or consideration for this series, but it does have a "money tree" sort of feel to it, if successive later activities are significantly more challenging than earlier ones, and continue becoming increasingly challenging at an increasing rate, to the point where a geometric progression does not feel appropriate. It would also offer the potential of "getting to" 1 000 000 under a praeternaturally unlikely series of events where in practice it may be very unlikely that the payout might even be as high as 30, 100 or 400.
That is all. No real point, just tickled me.
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