A165513 -id:A165513 - cp's OEIS frontend

A249351 Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Oct 26 2014

Here T(n,k) is defined to be the "k-th width" of the symmetric representation of sigma(n), with n>=1 and 1<=k<=2n-1. Explanation: consider the diagram of the symmetric representation of sigma(n) described in A236104, A237593 and other related sequences. Imagine that the diagram for sigma(n) contains 2n-1 equidistant segments which are parallel to the main diagonal [(0,0),(n,n)] of the quadrant. The segments are located on the diagonal of the cells. The distance between two parallel segment is equal to sqrt(2)/2. T(n,k) is the length of the k-th segment divided by sqrt(2). Note that the triangle contains nonnegative terms because for some n the value of some widths is equal to zero. For an illustration of some widths see Hartmut F. W. Hoft's contribution in the Links section of A237270.
Row n has length 2*n-1.
Row sums give A000203.
If n is a power of 2 then all terms of row n are 1's.
If n is an even perfect number then all terms of row n are 1's except the middle term which is 2.
If n is an odd prime then row n lists (n+1)/2 1's, n-2 zeros, (n+1)/2 1's.
The number of blocks of positive terms in row n gives A237271(n).
The sum of the k-th block of positive terms in row n gives A237270(n,k).
It appears that the middle diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).
It appears that the trapezoidal numbers (A165513) are also the numbers k > 1 with the property that some of the noncentral widths of the symmetric representation of sigma(k) are not equal to 1. - Omar E. Pol, Mar 04 2023

			Triangle begins:
  1;
  1,1,1;
  1,1,0,1,1;
  1,1,1,1,1,1,1;
  1,1,1,0,0,0,1,1,1;
  1,1,1,1,1,2,1,1,1,1,1;
  1,1,1,1,0,0,0,0,0,1,1,1,1;
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
  1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;
  1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;
  1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;
  1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1;
  ...
---------------------------------------------------------------------------
.        Written as an isosceles triangle              Diagram of
.              the sequence begins:               the symmetry of sigma
---------------------------------------------------------------------------
.                                                _ _ _ _ _ _ _ _ _ _ _ _
.                      1;                       |_| | | | | | | | | | | |
.                    1,1,1;                     |_ _|_| | | | | | | | | |
.                  1,1,0,1,1;                   |_ _|  _|_| | | | | | | |
.                1,1,1,1,1,1,1;                 |_ _ _|    _|_| | | | | |
.              1,1,1,0,0,0,1,1,1;               |_ _ _|  _|  _ _|_| | | |
.            1,1,1,1,1,2,1,1,1,1,1;             |_ _ _ _|  _| |  _ _|_| |
.          1,1,1,1,0,0,0,0,0,1,1,1,1;           |_ _ _ _| |_ _|_|    _ _|
.        1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;         |_ _ _ _ _|  _|     |
.      1,1,1,1,1,0,0,1,1,1,0,0,1,1,1,1,1;       |_ _ _ _ _| |      _|
.    1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1;     |_ _ _ _ _ _|  _ _|
.  1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1;   |_ _ _ _ _ _| |
.1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,1,1,1,1,1; |_ _ _ _ _ _ _|
...
From _Omar E. Pol_, Nov 22 2020: (Start)
Also consider the infinite double-staircases diagram defined in A335616.
For n = 15 the diagram with first 15 levels looks like this:
.
Level                         "Double-staircases" diagram
.                                          _
1                                        _|1|_
2                                      _|1 _ 1|_
3                                    _|1  |1|  1|_
4                                  _|1   _| |_   1|_
5                                _|1    |1 _ 1|    1|_
6                              _|1     _| |1| |_     1|_
7                            _|1      |1  | |  1|      1|_
8                          _|1       _|  _| |_  |_       1|_
9                        _|1        |1  |1 _ 1|  1|        1|_
10                     _|1         _|   | |1| |   |_         1|_
11                   _|1          |1   _| | | |_   1|          1|_
12                 _|1           _|   |1  | |  1|   |_           1|_
13               _|1            |1    |  _| |_  |    1|            1|_
14             _|1             _|    _| |1 _ 1| |_    |_             1|_
15            |1              |1    |1  | |1| |  1|    1|              1|
.
Starting from A196020 and after the algorithm described in A280850 and A296508 applied to the above diagram we have a new diagram as shown below:
.
Level                             "Ziggurat" diagram
.                                          _
6                                         |1|
7                            _            | |            _
8                          _|1|          _| |_          |1|_
9                        _|1  |         |1   1|         |  1|_
10                     _|1    |         |     |         |    1|_
11                   _|1      |        _|     |_        |      1|_
12                 _|1        |       |1       1|       |        1|_
13               _|1          |       |         |       |          1|_
14             _|1            |      _|    _    |_      |            1|_
15            |1              |     |1    |1|    1|     |              1|
.
The 15th row
of this seq:  [1,1,1,1,1,1,1,1,0,0,0,1,1,1,2,1,1,1,0,0,0,1,1,1,1,1,1,1,1]
The 15th row
of A237270:   [              8,            8,            8              ]
The 15th row
of A296508:   [              8,      7,    1,    0,      8              ]
The 15th row
of A280851    [              8,      7,    1,            8              ]
.
The number of horizontal steps (or 1's) in the successive columns of the above diagram gives the 15th row of this triangle.
For more information about the parts of the symmetric representation of sigma(n) see A237270. For more information about the subparts see A239387, A296508, A280851.
More generally, it appears there is the same correspondence between the original diagram of the symmetric representation of sigma(n) and the "Ziggurat" diagram of n. (End)

Index entries for sequences related to sigma(n)

Cf. A000203, A003056, A067742, A071562, A165513, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A245092, A245685, A246955, A246956, A247687, A249223, A250068, A250070, A250071, A262626, A280850, A280851, A296508, A235616, A347186.

(* function segments are defined in A237270 *)
a249351[n_] := Flatten[Map[segments, Range[n]]]
a249351[10] (* Hartmut F. W. Hoft, Jul 20 2022 *)

A214602 Integer areas of trapezoids such that all sides also have integer lengths.

Original entry on oeis.org

9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106

Offset: 1

Alonso del Arte, Jul 22 2012

By "trapezoid" here is meant a quadrilateral with exactly one pair of parallel sides.
If from an isosceles trapezoid having all sides with integer lengths we remove the widest rectangle having the same height as the trapezoid, we are left with two triangles that both correspond to the same Pythagorean triple.
Another possibility if we can remove a rectangle with the same width as the top of the trapezoid is that the remaining two triangles will correspond to two different Pythagorean triples both having the same smallest term, e.g., (15, 20, 25) and (15, 30, 36); this trapezoid has a base 51 units long, a top 1 unit long, height 15 units, left side 25 units and right side 36 units.
The smallest term that corresponds to more than one trapezoid is 15, which can be the area of a right trapezoid with a base measuring 7 units, a top of 3 units, height and left (or right) side 3 units, and right (or left) side 5 units; or an isosceles trapezoid with a base 9 units, top 1 unit, height 3 units, and left and right sides 5 units each.
The smallest term that is not congruent to 0, 2, 3 or 4 mod 6 (A047229) is 35. - Alonso del Arte, Aug 01 2012
Andrew Weimholt has pointed out that it is also possible to construct a trapezoid with the requirements above from which a rectangle can't be removed to leave two right triangles: one way to do this is to join two triangles corresponding to two different Pythagorean triples and then remove a parallelogram with two sides each measuring one less than the smallest number in the smaller Pythagorean triple. See Weimholt's illustration. - Alonso del Arte, Aug 06 2012

			21 is in the sequence because it is the area of a trapezoid with a base measuring 11 units, a top of 3 units, and left and right sides of 5 units each.

Andrew Weimholt, Illustration of a trapezoid with parallelograms removed

Cf. A165513, trapezoidal numbers.

Missing terms pointed out by Charles R Greathouse IV, Aug 02 2012, and Andrew Weimholt, Aug 06 2012

A165514 The complement of the trapezoidal numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 16, 28, 32, 64, 128, 136, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 1

Ant King, Sep 23 2009

nonn

Trapezoidal numbers (A165513) are polite numbers (A138591) that have a runsum representation which excludes one, and hence that can be depicted graphically by a trapezoid. This sequence is their complement, and Jones and Lord have shown that it is constructed from the powers of 2 (A000079), the perfect numbers (A000396) and those integers of the form 2^(k-1)*(2^k+1) where k is necessarily a power of 2 and 2^k + 1 is a Fermat prime (A019434).
Starting with 4, composite numbers (A002808) not a difference of non-neighboring triangular numbers (A000217). For T(x) - T(y), x - y > 1, where T are the triangular numbers, all other composite numbers can be represented as a triangular number difference. - Ed Pegg Jr, Feb 23 2016
It appears that these are also the numbers k with the property that all noncentral widths of the symmetric representation of sigma(k) are 1's, with a(1) = 1. Omar E. Pol, Mar 04 2023

			As the fifth integer which does not have a runsum representation which excludes one is 6, then a(5)=6.

Chai Wah Wu, Table of n, a(n) for n = 1..2577
Chris Jones and Nick Lord, Characterizing Non-Trapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.
Ron Knott, Introducing Runsums
Jim Smith, Trapezoidal numbers, Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 46-47.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A138591, A165513, A019434, A000396, A000079, A000217, A002808, A237593.

trapezoidal[n_] := Module[{result}, result = {}; Do[sum = 0; start = i; lis = {}; m = i; While[sum < n, sum = sum + m; lis = AppendTo[lis, m]; If[sum == n, AppendTo[result, lis]]; m++], {i, 2, Floor[n/2]}]; result]; Select[Range[10000], trapezoidal[#] == {} &]

More terms from Amiram Eldar, Aug 12 2019

A308468 "Trapezoidal numbers": numbers k such that the integers from 1 to k can be arranged in a trapezoid of H lines containing respectively L, L-1, L-2, ..., L-H+1 numbers from top to bottom. The rule is that from the second line, each integer is equal to the absolute value of the difference between the two numbers above it.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 48, 49, 51, 53

Offset: 1

Bernard Schott, May 29 2019

These numbers are called "nombres trapéziens" in French.
Some results from the article by "Diophante" (problème A352):
The powers of 2 are not trapezoidal.
Every odd number >= 3 is trapezoidal. In the case of k = 2m+1, a pattern can always be obtained with a trapezoid of height H = 2. The first line has the m+1 odd integers and the second the m even integers decreasing from 2m to 2, with this following arrangement:
1      2m+1        3        2m-1         5        ...
   2m       2m-2      2m-4        2m-6      ...          2
If H = L, the trapezoid becomes a triangle (examples for 3, 6 and 10 that are triangular numbers but 28 is not in trapezoid).
When an integer is trapezoidal, the number of ways for this to happen varies greatly; up to 30, the number of distinct solutions is greater when k is multiple of 6. Two symmetric trapezoids are considered to be identical.
It is not known if this sequence has a finite number of even terms.
If 34 is trapezoidal then the only possible trapezoid is necessarily of the form L = 10 and H = 4, and,
if 36 is trapezoidal, there are only two possible trapezoid forms, the first has L = 8 and H = 8 (it is a triangle) and the second one has L = 13 and H = 3.
Not to be confused with another definition of trapezoidal numbers, A165513. - N. J. A. Sloane, Jul 13 2019

			for k = 9:       1     9     3     7     5
                    8     6     4     2
------------------------------------------------------
for k = 10:      8     1     10     6
                    7     9      4
                       2      5
                          3

"Diophante", A352. Les nombres trapéziens, Sep. 2014 (in French).
Bert Dobbelaere, C++ program
Bernard Schott, Examples of numbers in trapezoid

Cf. A165513.

a(25)-a(37) from Bert Dobbelaere, Jul 14 2019

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A249351 Triangle read by rows in which row n lists the widths of the symmetric representation of sigma(n).

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A214602 Integer areas of trapezoids such that all sides also have integer lengths.

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A165514 The complement of the trapezoidal numbers.

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