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A335152 Number of vertices in polytope representing the number n.

%I A335152 #15 Jul 09 2025 04:52:58
%S A335152 1,2,3,3,4,5,6,6,5,6,7,8,9,10,11,10,11,12,13,13,14,15,16,17,15,16,15,
%T A335152 15,16,17,18,17,18,19,20,20,21,22,23,24,25,26,27,27,28,29,30,30,26,26,
%U A335152 27,27,28,29,30,31,32,33,34,35,36,37,38,36,37,38,39,39,40,41,42,42,43,44,44,44,45,46,47,48,45,46,47,48,49,50,51,51,52,53,54,54,55,56,57,58,59,58,57,55,56,57,58,58,59,60,61,62,63,64,65,66,67,68,69,69,68,69,70,71,66,67,68,68,66,67,68,65,66,67,68,69,70,71,72
%N A335152 Number of vertices in polytope representing the number n.
%C A335152 More than the usual number of terms are shown here in order to distinguish this sequence from A064047. The two sequences first differ at n=128.
%C A335152 Comments from _Ya-Ping Lu_, May 25 2020 (Start):
%C A335152 Concerning the sequences A064047 and the number of vertices of the polytope representing n (the present sequence). These two sequences are similar but not exactly the same.
%C A335152 As you can see from the pdf file attached, for n<=127, A064047 is the same as N_vert. For n > =128, A064047 is always greater than or equal to N_vert. This is due to the fact that in some cases not all the non-vertex numbers on the polytope can be written as the geometric mean of two integers on the polytope. See also A334810 and A334897.
%C A335152 The cases where A064047 is greater than N_vert are highlighted in yellow in the attached file. (End)
%H A335152 Ya-Ping Lu and Shu-Fang Deng, <a href="http://arxiv.org/abs/2003.08968">Properties of Polytopes Representing Natural Numbers</a>, arXiv:2003.08968 [math.GM], 2020.
%H A335152 Ya-Ping Lu, <a href="/A335152/a335152.pdf">Table comparing A064047 (column 2) and a(n) (column 3)</a>. [Yellow lines show where the values are different.]
%Y A335152 Cf. A064047, A333524, A334810, A334897.
%K A335152 nonn
%O A335152 1,2
%A A335152 _N. J. A. Sloane_, May 25 2020, based on correspondence with _Ya-Ping Lu_