A280223 -id:A280223 - cp's OEIS frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

0, 1, 2, 3, 4, 4, 6, 7, 8, 6, 10, 12, 12, 8, 14, 15, 16, 13, 18, 18, 20, 12, 22, 28, 24, 14, 26, 24, 28, 24, 30, 31, 32, 18, 34, 39, 36, 20, 38, 42, 40, 32, 42, 36, 44, 24, 46, 60, 48, 31, 50, 42, 52, 40, 54, 56, 56, 30, 58, 72, 60, 32, 62, 63, 64, 48

Offset: 0

Omar E. Pol, Jul 15 2014

nonn

Consider an irregular stepped pyramid with n steps. The base of the pyramid is equal to the symmetric representation of A024916(n), the sum of all divisors of all positive integers <= n. Two of the faces of the pyramid are the same as the representation of the n-th triangular numbers as a staircase. The total area of the pyramid is equal to 2*A024916(n) + A046092(n). The volume is equal to A175254(n). By definition a(2n-1) is A000203(n), the sum of divisors of n. Starting from the top a(2n-1) is also the total area of the horizontal part of the n-th step of the pyramid. By definition, a(2n) = A005843(n) = 2n. Starting from the top, a(2n) is also the total area of the irregular vertical part of the n-th step of the pyramid.
On the other hand the sequence also has a symmetric representation in two dimensions, see Example.
From Omar E. Pol, Dec 31 2016: (Start)
We can find the pyramid after the following sequences: A196020 --> A236104 --> A235791 --> A237591 --> A237593.
The structure of this infinite pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see the links).
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1, hence the sum of the areas of the terraces at the m-th level equals A000203(m).
Note that the stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the pyramid see A237593 and all its related sequences. (End)

			Illustration of initial terms:
----------------------------------------------------------------------
a(n)                             Diagram
----------------------------------------------------------------------
0    _
1   |_|\ _
2    \ _| |\ _
3     |_ _| | |\ _
4      \ _ _|_| | |\ _
4       |_ _|  _| | | |\ _
6        \ _ _|  _| | | | |\ _
7         |_ _ _|  _|_| | | | |\ _
8          \ _ _ _|  _ _| | | | | |\ _
6           |_ _ _| |    _| | | | | | |\ _
10           \ _ _ _|  _|  _|_| | | | | | |\ _
12            |_ _ _ _|  _|  _ _| | | | | | | |\ _
12             \ _ _ _ _|  _|  _ _| | | | | | | | |\ _
8               |_ _ _ _| |  _|  _ _|_| | | | | | | | |\ _
14               \ _ _ _ _| |  _| |  _ _| | | | | | | | | |\ _
15                |_ _ _ _ _| |_ _| |  _ _| | | | | | | | | | |\ _
16                 \ _ _ _ _ _|  _ _|_|  _ _|_| | | | | | | | | | |\
13                  |_ _ _ _ _| |  _|  _|  _ _ _| | | | | | | | | | |
18                   \ _ _ _ _ _| |  _|  _|    _ _| | | | | | | | | |
18                    |_ _ _ _ _ _| |  _|     |  _ _|_| | | | | | | |
20                     \ _ _ _ _ _ _| |      _| |  _ _ _| | | | | | |
12                      |_ _ _ _ _ _| |  _ _|  _| |  _ _ _| | | | | |
22                       \ _ _ _ _ _ _| |  _ _|  _|_|  _ _ _|_| | | |
28                        |_ _ _ _ _ _ _| |  _ _|  _ _| |  _ _ _| | |
24                         \ _ _ _ _ _ _ _| |  _| |    _| |  _ _ _| |
14                          |_ _ _ _ _ _ _| | |  _|  _|  _| |  _ _ _|
26                           \ _ _ _ _ _ _ _| | |_ _|  _|  _| |
24                            |_ _ _ _ _ _ _ _| |  _ _|  _|  _|
28                             \ _ _ _ _ _ _ _ _| |  _ _|  _|
24                              |_ _ _ _ _ _ _ _| | |  _ _|
30                               \ _ _ _ _ _ _ _ _| | |
31                                |_ _ _ _ _ _ _ _ _| |
32                                 \ _ _ _ _ _ _ _ _ _|
...
a(n) is the total area of the n-th set of symmetric regions in the diagram.
.
From _Omar E. Pol_, Aug 21 2015: (Start)
The above structure contains a hidden pattern, simpler, as shown below:
Level                              _ _
1                                _| | |_
2                              _|  _|_  |_
3                            _|   | | |   |_
4                          _|    _| | |_    |_
5                        _|     |  _|_  |     |_
6                      _|      _| | | | |_      |_
7                    _|       |   | | |   |       |_
8                  _|        _|  _| | |_  |_        |_
9                _|         |   |  _|_  |   |         |_
10             _|          _|   | | | | |   |_          |_
11           _|           |    _| | | | |_    |           |_
12         _|            _|   |   | | |   |   |_            |_
13       _|             |     |  _| | |_  |     |             |_
14     _|              _|    _| |  _|_  | |_    |_              |_
15   _|               |     |   | | | | |   |     |               |_
16  |                 |     |   | | | | |   |     |                 |
...
The symmetric pattern emerges from the front view of the stepped pyramid.
Note that starting from this diagram A000203 is obtained as follows:
In the pyramid the area of the k-th vertical region in the n-th level on the front view is equal to A237593(n,k), and the sum of all areas of the vertical regions in the n-th level on the front view is equal to 2n.
The area of the k-th horizontal region in the n-th level is equal to A237270(n,k), and the sum of all areas of the horizontal regions in the n-th level is equal to sigma(n) = A000203(n). (End)
From _Omar E. Pol_, Dec 31 2016: (Start)
Illustration of the top view of the pyramid with 16 levels:
.
n   A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1      1   =      1      |_| | | | | | | | | | | | | | | |
2      3   =      3      |_ _|_| | | | | | | | | | | | | |
3      4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
4      7   =      7      |_ _ _|    _|_| | | | | | | | | |
5      6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
6     12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
7      8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
8     15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
9     13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
16    31   =     31      |_ _ _ _ _ _ _ _ _|
... (End)

Robert Price, Table of n, a(n) for n = 0..20000
Omar E. Pol, A pyramid related to the divisor function and other integers sequences
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Cf. A000203, A004125, A024916, A005843, A175254, A196020, A224880, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244970, A244971, A245093, A261350, A262626, A277437, A279387, A280223, A280295.

Table[If[EvenQ@ n, n, DivisorSigma[1, (n + 1)/2]], {n, 0, 65}] (* or *)
Transpose@ {Range[0, #, 2], DivisorSigma[1, #] & /@ Range[#/2 + 1]} &@ 65 // Flatten (* Michael De Vlieger, Dec 31 2016 *)
With[{nn=70},Riffle[Range[0,nn,2],DivisorSigma[1,Range[nn/2]]]] (* Harvey P. Dale, Aug 05 2024 *)

a(2*n-1) + a(2n) = A224880(n).

A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 3, 1, 2, 2, 4, 1, 3, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 2, 4, 2, 5, 3, 1, 3, 4, 2, 1, 5, 2, 1, 1, 4, 4, 2, 2, 5, 3, 1, 5, 2, 2, 2, 3, 5, 3, 1, 6, 3, 1, 2, 4, 2, 3, 3, 1, 1, 6, 4, 2, 5, 3, 2, 3, 1, 2, 2, 4, 4, 1, 1, 6, 4, 1, 3, 1, 3

Offset: 1

Omar E. Pol, Jan 10 2017

nonn

Descending by the main diagonal of the pyramid, A071562 gives the levels where we can find a terrace.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
a(n) is the length of the n-th vertical line segment at the main diagonal of the pyramid.
a(n) is the precipice of A071562(n).
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
Equals nonzero terms of A259179. - Omar E. Pol, Apr 17 2018

Robert Price, Table of n, a(n) for n = 1..13750
Omar E. Pol, Illustration of the isosceles triangle of A237593 (rows 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

For more information about the precipices see A276112, A277437, A280223 and A280295.

Cf. A000203, A071562, A196020, A235791, A236104, A237048, A237591, A237593, A237270, A237271, A244050, A245092, A259179, A262626.

Differences@ Select[Range@ 228, Function[n, Total@ Select[Divisors@ n, Sqrt[n/2] <= # < Sqrt[2 n] &] != 0]] (* Michael De Vlieger, Jan 13 2017, after Robert G. Wilson v at A071562 *)

a(n) = A280223(A071562(n)).

More terms from Michael De Vlieger, Jan 13 2017

A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

Original entry on oeis.org

1, 3, 2, 5, 4, 9, 7, 6, 12, 20, 8, 10, 21, 36, 72, 11, 13, 25, 50, 91, 144, 14, 16, 32, 56, 112

Offset: 1

Omar E. Pol, Dec 29 2016

This is a permutation of the natural numbers.
Column k lists the numbers with precipice k. For more information about the precipices see A280223 and A280295.
The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the m-th level of the pyramid are also the parts of the symmetric representation of sigma(m), m >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
If a number m is in the column k and k > 1 then m + 1 is the column k - 1.
The largest Dyck path of the symmetric representations of next k - 1 positive integers greater than T(n,k) shares the middle point of the largest Dyck path of the symmetric representation of sigma(T(n,k)). For more information see A237593.

			The corner of the square array begins:
   1,  2,  9, 20, 72, 144,
   3,  4, 12, 36, 91,
   5,  6, 21, 50,
   7, 10, 25,
   8, 13,
  11,
  ...
T(1,6) = 144 because it is the smallest number with precipice 6.

Row 1 gives A280295.
Column 1 gives A276112.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223.

T(n,1) = A071562(n+1) - 1.

a(20)-a(26) from Omar E. Pol, Jan 02 2017

A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

Original entry on oeis.org

1, 2, 9, 20, 72, 144

Offset: 1

Omar E. Pol, Dec 31 2016

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k), k >= 1.
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437 and A280223.
Is this sequence infinite?

			a(3) = 9 because descending by the main diagonal of the pyramid, the height difference between the level 9 and the level of the next terrace is equal to 3, and 9 is the smallest number with this property.

Omar E. Pol, Perspective view of the pyramid (first 16 steps)

Row 1 of A277437.

Cf. A000203, A196020, A235791, A236104, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A276112, A277437, A279286, A279385, A280223.

a(6) from Omar E. Pol, Jan 02 2017

A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

Original entry on oeis.org

1, 3, 5, 7, 8, 11, 14, 15, 17, 19, 23, 24, 27, 29, 31, 34, 35, 39, 41, 44, 47, 48, 49, 53, 55, 59, 62, 63, 65, 69, 71, 76, 79, 80, 83, 87, 89, 90, 95, 97, 98, 99, 103, 107, 109, 111, 116, 119, 120, 125, 127, 129, 131, 134, 139, 142, 143, 149, 152, 153, 155, 159

Offset: 1

Omar E. Pol, Jan 02 2017

nonn

The structure of the stepped pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593.
The terraces at the k-th level of the pyramid are also the parts of the symmetric representation of sigma(k).
The stepped pyramid is also one of the 3D-quadrants of the stepped pyramid described in A244050.
For more information about the precipices see A277437, A280223 and A280295.
From Hartmut F. W. Hoft, Feb 02 2022: (Start)
Also partial sums of A280919.
a(n) is also the largest number of a Dyck path that crosses the diagonal at point A282131(n) which also is the rightmost number in each nonzero row of the irregular triangle in A279385. (End)

			From _Hartmut F. W. Hoft_, Feb 02 2022: (Start)
      n: 1  2  3  4  5  6  7  8  9 10 11 12 13 14 index.
A282131: 1  2  3  5  6  7  9 11 12 13 15 17 18 20 position on diagonal.
A276112: 1  3  5  7  8 11 14 15 17 19 23 24 27 29 max index of Dyck path.
A280919: 1  2  2  2  1  3  3  1  2  2  4  1  3  2 paths at diag position.
(End)

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Column 1 of A277437.
Cf. A000203, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244050, A245092, A259179, A262626, A279286, A279385, A280223, A280295.
Cf. A280919, A282131.

(* last computed value of a280919[ ] is dropped to avoid a potential undercount of crossings *)
a240542[n_] := Sum[(-1)^(k+1)Ceiling[(n+1)/k-(k+1)/2], {k, 1, Floor[-1/2+1/2 Sqrt[8n+1]]}]
a280919[n_] := Most[Map[Length, Split[Map[a240542, Range[n]]]]]
A276112[160] (* Hartmut F. W. Hoft, Feb 02 2022 *)

a(n) = A071562(n+1) - 1.

a(n) = Sum_{i=1..n} A280919(i), n >= 1. - Hartmut F. W. Hoft, Feb 02 2022

A281012 Zig-zag path that we can find descending by the main diagonal of the pyramid described in A245092.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 3, 2, 3, 2, 1, 1, 2, 1, 2, 2, 4, 2, 1, 1, 3, 2, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 2, 2, 3, 2, 3, 2, 1, 1, 1, 1, 4, 2, 2, 2, 4, 2, 3, 2, 1, 1, 2, 2, 4, 2, 2, 3, 5, 2, 3, 2, 1, 1, 3, 2, 4, 2, 2, 2, 1, 2, 5, 2, 2, 1, 1, 2, 1, 1, 4, 2, 4, 2, 2, 2, 2, 2, 5, 2, 3, 4, 1, 1, 5, 2, 2, 1, 2

Offset: 1

Omar E. Pol, Feb 11 2017

nonn

The odd-indexed terms are the widths of the successive terraces in the main diagonal of the pyramid.
Conjecture: the odd-indexed terms give A281007, the positive terms in A067742.
The even-indexed terms are the precipices from the successive terraces, descending by the main diagonal of the pyramid.
The even-indexed terms give A280919, the first differences of A071562.
The structure of the pyramid arises after the 90-degree-zig-zag folding of the diagram of the isosceles triangle A237593 (see links).
The mentioned pyramid is also a 3D-quadrant of the pyramid described in A244050.

Omar E. Pol, A pyramid related to the sum-of-divisors function and other integers sequences
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Also A281007 and A280919 interleaved (conjectured).

Cf. A067742, A071562, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A279387, A280223.

A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

Original entry on oeis.org

1, 7, 13, 0, 20, 15, 43, 0, 66, 0, 24, 49, 59, 0, 134, 0, 60, 113, 0, 86, 0, 104, 165, 0, 48, 245, 0, 132, 0, 224, 0, 198, 0, 124, 57, 317, 0, 192, 0, 350, 0, 326, 0, 104, 211, 0, 434, 0, 216, 0, 0, 647, 0, 344, 0, 186, 331, 0, 584, 0, 270, 0, 234, 0, 672, 0, 350, 171, 0, 156, 639, 0, 672, 0, 390, 0, 368, 0, 956

Offset: 1

Omar E. Pol, Feb 19 2018

nonn

Row sums of A299693.

Cf. A196020, A235791, A236104, A237048, A237591, A237593, A240542, A245092, A259179, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

Original entry on oeis.org

1, 3, 4, 7, 6, 0, 12, 8, 15, 13, 18, 12, 0, 28, 14, 24, 0, 24, 31, 18, 39, 20, 0, 42, 32, 36, 24, 0, 60, 31, 42, 40, 0, 56, 30, 0, 72, 32, 63, 48, 54, 0, 48, 91, 38, 60, 56, 0, 90, 42, 0, 96, 44, 84, 0, 78, 72, 48, 0, 124, 57, 93, 72, 98, 54, 0, 120, 72, 0, 120, 80, 90, 60, 0, 168, 62, 96, 0, 104, 127, 84, 0

Offset: 1

Omar E. Pol, Feb 19 2018

			Triangle begins:
   1;
   3,  4;
   7,  6;
   0;
  12,  8;
  15;
  13, 18, 12;
   0;
  28, 14, 24;
   0;
  24;
  31, 18;
  39, 20;
   0;
  42, 32, 36, 24;
   0;
...

Nonzero terms give A000203.
Row sums give A299472.
Cf. A259179(n) is the number of positive terms in row n.
Cf. A071562, A196020, A235791, A236104, A237048, A237591, A237593, A240542, A244050, A245092, A276112, A277437, A279286, A279385, A280919, A280223, A282131, A282197, A280295, A281012.

T(n,m) = A000203(A279385(n,m)) if A279385(n,m) > 0, otherwise T(n,m) = 0.

cp's OEIS Frontend

A245092 The even numbers (A005843) and the values of sigma function (A000203) interleaved.

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A280919 Precipices from the successive terraces, descending by the main diagonal of the pyramid described in A245092. Also first differences of A071562.

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A277437 Square array read by antidiagonals upwards in which T(n,k) is the n-th number j such that, descending by the main diagonal of the pyramid described in A245092, the height difference between the level j (starting from the top) and the level of the next terrace is equal to k.

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A280295 Smallest number with precipice n. Descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to n.

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A276112 Numbers with precipice 1: descending by the main diagonal of the pyramid described in A245092, the height difference between the level a(n) (starting from the top) and the level of the next terrace is equal to 1.

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A281012 Zig-zag path that we can find descending by the main diagonal of the pyramid described in A245092.

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A299472 a(n) is the sum of all divisors of all numbers k whose associated largest Dyck path contains the point (n,n) in the diagram of the symmetric representation of sigma(k) described in A237593, or 0 if no such k exists.

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A299693 Irregular triangle read by rows in which row n lists the total sum of the divisors of all numbers k such that the largest Dyck path of the symmetric representation of sigma(k) contains the point (n,n); or row n is 0 if no such k exists.

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