COMMENTS
Natural numbers x that cannot be expressed as floor(y)+ceiling(y)+round(y) where y = log(x). - Bill McEachen, Oct 25 2023
Discussion
Thu Oct 26
07:47
Bill McEachen: @Kevin Let's just drop it then, I always have too many in the queue to fiddle with such. Thanks for the input...Can you just revert then please ?
08:46
Kevin Ryde: Undo button.
Discussion
Wed Oct 25
11:47
Andrew Howroyd: Is this proven? It seems slightly odd that floor(y)+ceiling(y)+round(y) where y = log(x) would have any kind of linear recurrence.
11:50
Andrew Howroyd: NM, this sequence is just multiples of 3.
20:03
Kevin Ryde: Something badly wrong in the variable names in the statement. If the assertion is that these are x != floor(y)+ceil(y)+round(y) where y=log(x), then that's every x>=6, which is not this sequence.
20:05
Kevin Ryde: With that fixed, the use of log() is an obscurity which serves only to exclude integer y.
20:28
Bill McEachen: @Kevin you lost me, give me a bit pls.
21:17
Bill McEachen: @Kevin Y is not integer, right? For contrast, via WA we can use floor(log10(x)) + ceiling(log10(x)) + round(log10(x)) to show that one can obtain values 1,2,3... ie all values with base 10 log. [all integer base logs w/ minor exceptions ?]
But if you think the comment useless or moot, by all means discard it. I thought to add it because I must have come across it.
23:13
Bill McEachen: I think the better wording is: Conjecture: natural numbers x that cannot be expressed as floor(y)+ceiling(y)+round(y) with y any non-integer base log of x, n>0. I had only considered integer base logs along with natl log, but scrutinized further upon your comment.
Thu Oct 26
01:56
Kevin Ryde: If y is an integer, then clearly floor(y) + ceil(y) + round(y) = 3*y.
02:01
Kevin Ryde: If y is not an integer, then floor(y) + ceil(y) + round(y) = 3*floor(y) + (1 or 2 depending on fractional part of y, umm < 1/2 or >= 1/2). The log() has no role except that natural logs of integers make both types of fractional part of y.
02:03
Kevin Ryde: At any rate, have to drop you back to editing since "numbers x" with "y = log(x)" is making variable name x do double-duty (or is outright wrong if read literally).
COMMENTS
Natural numbers x that cannot be expressed as floor(y)+ceiling(y)+round(y) where y = log(x). = _- _Bill McEachen_, Oct 25 2023
COMMENTS
Natural numbers x that cannot be expressed as floor(y)+ceiling(y)+round(y) where y = log(x). = Bill McEachen, Oct 25 2023
Discussion
Wed Oct 25
07:24
Bill McEachen: eg x=2 thru 13 yields 2,4,4,5,5,5,7,7,7,7,7,8,...etc
