[seqfan] Re: One more conjecture for n-gonal numbers

[Jim Nastos]

If we include "n" as an "n-gonal" number (like 8 is an 8-gonal number
and not a k-gonal number for any k less than 8) then the sequence:

"Smallest n-gonal number which is not a k-gonal number for any k < n"
for n=7,8,9,.... is
7, 8, 24, 27, 11, 33, 13, 14, 42, 88, 17, 165, 19, 20, 60, 63, 23, 69,
72, 26, 255, 160, 29, 87, ...


And if we instead want the smallest n-gonal number, not including n,
which is not a k-gonal number for any k less than n, we get:

"Smallest n-gonal number larger than n which is not a k-gonal number
for any k < n" for n=7,8,9,...
18, 40, 24, 27, 30, 33, 115, 39, 42, 88, 48, 165, 54, 57, 60, 63, 130,
69, 72, 245, 255, 160, 84, 87, ...


If your conjecture is false, then one (I'm not sure which) or both of
these sequences is not well-defined.

If anyone wants to confirm these values, please feel free to go ahead
and author them as well.

J



On Thu, Apr 29, 2010 at 11:03 AM, Vladimir Shevelev <shevelev at bgu.ac.il> wrote:
> Conjecture. For every n>=4, except of n=6, there exists a n-gonal number N wich is not k-gonal  for 3<=k<n.
> In case of n=6 it is easy to prove that every hexagonal number is also triangular, i.e. N does not exist.
>
> Best regards,
> Vladimir
>
>  Shevelev Vladimir‎
>
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