In One Sequence Or Another (counter-intuitive?)
Leroy Quet
qq-quet at mindspring.com
Thu Mar 25 19:21:27 CET 2004
I wrote in part:
>...
>So, if we were to label, in order starting from the square nearest the
>origin, the squares in the grid-path with 1, 2, 3, 4, 5, 6,...
>
>then all integers of the form
>
>floor(f(k)) + k
>
>would be at the top-most squares of each vertical section of the
>grid-path;
>
>and all integers of the form
>
>ceiling(g(k) + k - 1
>
>would be a the right-most squares of each horizontal section of the
>grid-path.
>
>(I am assumming that the path is tending up and to the right.)
>
>Since every square in the path is either a right-most square or an
>upper-most square, but not both, every positive integer is either
>
>floor(f(k)) +k
>or
>ceiling(g(k)) +k-1.
>
>(Unrigorous, yes.)
>...
I must clarify that, by "vertical section" and "horizontal section" the
section could consist of only one square.
I must give a visual example to help explain what I mean, I bet.
Below I give the grid path where
floor(f(k)) + k -> primes,
and ceiling(g(k))+ k -1 -> 1 and composites.
(use fixed-width font.)
# # # # # #17
# # # # # #16
# # # # # #15
# # # # #1314
# # # #1112 #
# # # #10 # #
# # # # 9 # #
# # # 7 8 # #
# # 5 6 # # #
2 3 4 # # # #
1 # # # # # #
Notice the locations of the primes vs the locations of the composites
(and 1).
thanks,
Leroy Quet
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