A334810 -id:A334810 - cp's OEIS frontend

A335152 Number of vertices in polytope representing the number n.

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, 15, 16, 17, 18, 17, 18, 19, 20, 20, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 30, 26, 26, 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

Offset: 1

N. J. A. Sloane, May 25 2020, based on correspondence with Ya-Ping Lu

nonn

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.
Comments from Ya-Ping Lu, May 25 2020 (Start):
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.
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.
The cases where A064047 is greater than N_vert are highlighted in yellow in the attached file. (End)

Ya-Ping Lu and Shu-Fang Deng, Properties of Polytopes Representing Natural Numbers, arXiv:2003.08968 [math.GM], 2020.
Ya-Ping Lu, Table comparing A064047 (column 2) and a(n) (column 3). [Yellow lines show where the values are different.]

Cf. A064047, A333524, A334810, A334897.

A143836 Triangle read by rows: T(r,c) = (prime(r+2) + prime(c+1))/2.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 8, 9, 10, 12, 10, 11, 12, 14, 15, 11, 12, 13, 15, 16, 18, 13, 14, 15, 17, 18, 20, 21, 16, 17, 18, 20, 21, 23, 24, 26, 17, 18, 19, 21, 22, 24, 25, 27, 30, 20, 21, 22, 24, 25, 27, 28, 30, 33, 34, 22, 23, 24, 26, 27, 29, 30, 32, 35, 36, 39, 23, 24, 25, 27, 28, 30, 31, 33, 36, 37, 40, 42

Offset: 1

Pierre CAMI, Sep 02 2008

The number of appearances of m >= 1 in this sequence is A061357(m). Conjecture: Every integer >= 4 appears at least once in this sequence. - Ya-Ping Lu, Mar 05 2023

The number of composites between 3 and (r+2)-th prime missing from Row 1 through Row r in the triangle is A334810(r+2). - Ya-Ping Lu, Mar 24 2023

			Triangle begins:
   4;
   5,  6;
   7,  8,  9;
   8,  9, 10, 12;
  10, 11, 12, 14, 15;
  ...

Cf. A098090 (1st column, except 1st term), A024675 (right diagonal).

T(r,c) = (prime(r+2) + prime(c+1))/2; \\ Michel Marcus, Mar 07 2023

Name simplified by Ya-Ping Lu, Mar 05 2023

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A335152 Number of vertices in polytope representing the number n.

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A143836 Triangle read by rows: T(r,c) = (prime(r+2) + prime(c+1))/2.

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