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

Thu Apr 29 22:24:35 CEST 2010

```I corrected some misprints.

Thanks, Jim!

Before formulation the conjecture, I already have sent a few terms of such sequence but it is not appeared in OEIS:

%I A176948
%S A176948 3,4,5,0,7,8,24,27,11,33,13,14,42,88,17,165,19,20,60,63,23,69,72,26
%N A176948 a(n) is the smallest solution of equation A176774(x)=n and a(n)=0 if this equation has not solution
%C A176948 If n is odd prime, then a(n)=n.
%Y A176948 A176774 A176744 A176747 A176775 A175873 A176874
%K A176948 nonn
%O A176948 3,1
%A A176948 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 29 2010

You gave more terms, and I ask you to extend this sequence.
The second your sequence is also interesting.

I also sent the sequence of composite numbers of A176948.

%I A176949
%S A176949 4,8,14,20,26,32,38,44,50,56,62,68,74,77,80
%N A176949 Composite numbers for which A176948(n)=n
%C A176949 If n>=3 is prime then A176948(n)=n. The sequence lists composite numbers with this property. It is interesing that there are many common first terms with A140164 (but, e.g.,77).
%Y A176949 A176948 A176774 A176744 A176747 A176775 A175873 A176874
%K A176949 nonn
%O A176949 3,1
%A A176949 Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 29 2010

----- Original Message -----
From: Jim Nastos <nastos at gmail.com>
Date: Thursday, April 29, 2010 23:00
Subject: [seqfan] Re: One more conjecture for n-gonal numbers
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 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,
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
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