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
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 lexico-
graphically earliest seq of distinct
terms with this property... but!
Here comes the (Math-)fun part.
I thought that S would increase monoto-
nically for ever when I bumped into
a(9) = 2298280 (and not, as above,
= 2298281).
I realized that S could decrease at
some point! Indeed, this 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, we have to backtrack
-- and increase by 1 the terms ending
in "0" that would stop the sequence;
this is what I did above.
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 stage to [x.y],
(the divisor), but I don't know if it
means something for the future of S].
Best,
É.
à+
É.
Catapulté de mon aPhone
