We Love Commas
« In music, the schisma is the interval between a Pythagorean comma and a syntonic comma. » (Wikipedia)
Here is my latest post to the Math-Fun mailing list (May 3rd, 2020):
Hello Math-Fun,
S = 2,5,26,159,1447,10274,45206,280278,2298281,...If we frame any comma above with its two closest digits we get 2.5 for the first comma, 5.2 for the second comma, 6.1 for the third one, etc.
Look now:
the division of 5 by 2 starts with 2.5
the division of 26 by 5 starts with 5.2
the division of 159 by 26 starts with 6.1
the division of 1447 by 159 starts with 9.1, etc.
[In other words, a(n+1)/a(n) starts with [x.y] with x = the rightmost digit of a(n) and y = the leftmost digit of a(n+1)].
As usual, we want S to be the lexicographically earliest seq of distinct terms with this property... but!
Here comes the (Math-)fun part...
I thought that S would increase monotonically for ever – when I bumped into a(9) = 2298280 [and not, as above, a(9) = 2298281].
I realized then that S could decrease at some point! Indeed, the following example works: 120,24,... as the result of 24/120 starts with 0.2
But this "zero comma" trick doesn't systematically work with integers ending in 0 (it works rarely, in reality). If it doesn't when you need it, you'll have to backtrack – which consists most of the time in increasing by 1 the term ending in "0" that would stop the sequence; this is what I did above with a(9).
The question remains: will the "+1" trick always work? Or will S stop at some point? Enter into a loop? Extend for ever?
[I've noticed that the ratio a(n+1)/a(n) gets closer at every step to [x.y], (the divisor), but I don't know if it means something for the future of S].
Best,
É.
____________________
Update, 19:00 Brussels time, same day
Jean-Marc Falcoz was quick to compute the hereunder 62-term list: merci beaucoup, Jean-Marc !
S = 2, 5, 26, 159, 1447, 10274, 45206, 280278, 2298281, 2757938, 22615092, 56537732, 118729239, 1080436075, 5942398412, 12479036665, 69882605324, 293506942361, 381559025069, 3548498933141, 4967898506397, 36265659096698, 297378404592924, 1219251458830989, 11095188275362001, 12204707102898201, 13425177813188022, 29535391189013652, 79745556210336864, 342905891704448512, 994427085942900864, 4375479178148764161, 7438314602852899842, 15620460665991090177, 110905270728536752128, 987056909483977080832, 2171525200864749813761, 2605830241037700300801, 3387579313349010653184, 13889075184730944569344, 63889745849762352988162, 134168466284500953858048, 1086764576904457704439808, 9672204734449674992222208, 85115401663157146803503104, 365996227151575708921233408, 3037768685358078631436353536, 18530388980684281191078035456, 113035372782174132198055084032, 248677820120783118983218855936, 1516934702736777191164183838721, 1668628173010454910280602222592, 3837844797924046969185329217536, 23794637747129090871179069095936, 145147290257487474130030681915392, 333838767592221246343705947799552, 968132426017441700865860094132224, 4259782674476743944978386256920576, 26410652581755813957663950781808641, 34333848356282558144963136016351232, 99568160233219433838956955257798656, 657149857539248344502789829023498241, ...
Commentaires
Enregistrer un commentaire