cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Sep 26 2015

Keywords

Comments

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

Examples

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   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      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Links

Crossrefs

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A346875 Irregular triangle read by rows in which row n lists the row A000384(n) of A237591, n >= 1.

Original entry on oeis.org

1, 4, 1, 1, 8, 3, 2, 1, 1, 15, 5, 3, 2, 1, 1, 1, 23, 8, 5, 2, 2, 2, 1, 1, 1, 34, 11, 6, 4, 3, 2, 2, 1, 1, 1, 1, 46, 16, 8, 5, 4, 2, 3, 1, 2, 1, 1, 1, 1, 61, 20, 11, 6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 77, 26, 14, 8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1, 96, 32, 16
Offset: 1

Views

Author

Omar E. Pol, Aug 06 2021

Keywords

Comments

The characteristic shape of the symmetric representation of sigma(A000384(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a valley and the largest Dyck path has a peak.
So knowing this we can know if a number is a hexagonal number (or not) just by looking at the diagram, even ignoring the concept of hexagonal number.
Therefore we can see a geometric pattern of the distribution of the hexagonal numbers in the stepped pyramid described in A245092.
T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000384(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000384(n-1).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th hexagonal number into exactly k + 1 consecutive parts.

Examples

			Triangle begins:
   1;
   4,  1,  1;
   8,  3,  2,  1, 1;
  15,  5,  3,  2, 1, 1, 1;
  23,  8,  5,  2, 2, 2, 1, 1, 1;
  34, 11,  6,  4, 3, 2, 2, 1, 1, 1, 1;
  46, 16,  8,  5, 4, 2, 3, 1, 2, 1, 1, 1, 1;
  61, 20, 11,  6, 5, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  77, 26, 14,  8, 5, 5, 3, 2, 3, 2, 1, 2, 1, 1, 1, 1, 1;
  96, 32, 16, 10, 7, 5, 4, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column H gives the nonzero hexagonal numbers (A000384).
Column S gives the sum of the divisors of the hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    H    S   Diagram
-------------------------------------------------------------------------
                 _         _                 _                         _
  1    1    1   |_|       | |               | |                       | |
                 1        | |               | |                       | |
                       _ _| |               | |                       | |
                      |    _|               | |                       | |
                 _ _ _|  _|                 | |                       | |
  2    6   12   |_ _ _ _| 1                 | |                       | |
                    4    1                  | |                       | |
                                       _ _ _|_|                       | |
                                   _ _| |                             | |
                                  |    _|                             | |
                                 _|  _|                               | |
                                |_ _|1 1                              | |
                                | 2                                   | |
                 _ _ _ _ _ _ _ _|4                           _ _ _ _ _| |
  3   15   24   |_ _ _ _ _ _ _ _|                           |  _ _ _ _ _|
                        8                                   | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|1 1
                                             _ _ _| | 1
                                            |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.

Crossrefs

Row sums give A000384, n >= 1.
Row lengths give A005408.
Column 1 is A267682, n >= 1.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A237591, A237593, A245092, A249351, A262626.

A346876 Irregular triangle read by rows in which row n is the "n-th even perfect number" row of A237591, n >= 1.

Original entry on oeis.org

4, 1, 1, 15, 5, 3, 2, 1, 1, 1, 249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4065, 1355, 678, 407, 271, 194, 146, 113, 91, 75, 62, 52, 45, 40, 34, 30, 27, 25, 22, 19, 19, 16, 15, 14, 13, 12, 12, 10, 10, 9, 9, 8, 8, 7
Offset: 1

Views

Author

Omar E. Pol, Aug 06 2021

Keywords

Comments

The characteristic shape of the symmetric representation of sigma(A000396(n)) consists in that the diagram has only one region (or part) and that region has whidth 1 except in the main diagonal where the width is 2.
So knowing this characteristic shape we can know if a number is an even perfect number (or not) just by looking at the diagram, even ignoring the concept of even perfect number (see the examples).
Therefore we can see a geometric pattern of the distribution of the even perfect numbers in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000396(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000396(n) assuming there are no odd perfect numbers.
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th even perfect number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th perfect number into exactly k + 1 consecutive parts.

Examples

			Triangle begins:
    4, 1, 1;
   15, 5, 3, 2, 1, 1,1;
  249,83,42,25,17,13,9,7,6,5,5,3,4,2,3,2,2,2,2,2,1,2,1,2,1,1,1,1,1,1,1;
...
Illustration of initial terms:
Column P gives the even perfect numbers (A000396 assuming there are no odd perfect numbers).
Column S gives A139256, the sum of the divisors of the even perfect numbers equals the area (and the number of cells) of the associated diagram.
-------------------------------------------------------------------------
  n    P   S    Diagram:   1                                           2
-------------------------------------------------------------------------
                           _                                           _
                          | |                                         | |
                          | |                                         | |
                       _ _| |                                         | |
                      |    _|                                         | |
                 _ _ _|  _|                                           | |
  1    6   12   |_ _ _ _| 1                                           | |
                    4    1                                            | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                                      | |
                                                             _ _ _ _ _| |
                                                            |  _ _ _ _ _|
                                                            | |
                                                         _ _| |
                                                     _ _|  _ _|
                                                    |    _|
                                                   _|  _|
                                                  |  _|1 1
                                             _ _ _| | 1
                                            |  _ _ _|2
                                            | |  3
                                            | |
                                            | |5
                 _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  2   28   56   |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                              15
.
For n = 3, P = 496, the diagram is too large to include here. To draw that diagram note that the lengths of the line segments of the smallest Dyck path are [248, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 248] and the lengths of the line segments of the largest Dyck path are [249, 83, 42, 25, 17, 13, 9, 7, 6, 5, 5, 3, 4, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 4, 3, 5, 5, 6, 7, 9, 13, 17, 25, 42, 83, 249].

Links

Crossrefs

Row sums give A000396.
Row lengths give A000668.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A139256, A237591, A237593, A245092, A249351, A262626.

Programs

  • PARI
    row235791(n) = vector((sqrtint(8*n+1)-1)\2, i, 1+(n-(i*(i+1)/2))\i);
row(n) = {my(orow = concat(row235791(n), 0)); vector(#orow -1, i, orow[i] - orow[i+1]); } \\ A237591
tabf(nn) = {for (n=1, nn, my(p=prime(n)); if (isprime(2^n-1), print(row(2^(n-1)*(2^n-1)));););}
tabf(7) \\ Michel Marcus, Aug 31 2021

Extensions

More terms from Michel Marcus, Aug 31 2021
Name edited by Michel Marcus, Jun 16 2023

A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

Original entry on oeis.org

2, 2, 1, 3, 2, 4, 2, 1, 6, 3, 1, 1, 7, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 2, 1, 12, 5, 2, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 6, 3, 2, 2, 1, 1, 19, 7, 4, 2, 2, 1, 1, 1, 21, 8, 4, 2, 2, 2, 1, 1, 22, 8, 4, 3, 2, 1, 2, 1, 24, 9, 4, 3, 2, 2, 1, 1, 1, 27, 10, 5, 3, 2, 2, 1, 2, 1
Offset: 1

Views

Author

Omar E. Pol, Aug 06 2021

Keywords

Comments

The characteristic shape of the symmetric representation of sigma(prime(n)) consists in that the diagram contains exactly two regions (or parts) and each region is a rectangle (or bar), except for the first prime number (the 2) whose symmetric representation of sigma(2) consists of only one region which contains three cells.
So knowing this characteristic shape we can know if a number is prime (or not) just by looking at the diagram, even ignoring the concept of prime number.
Therefore we can see a geometric pattern of the exact distribution of prime numbers in the stepped pyramid described in A245092.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(prime(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000040(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th prime into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th prime into exactly k + 1 consecutive parts.

Examples

			Triangle begins:
   2;
   2, 1;
   3, 2;
   4, 2, 1;
   6, 3, 1, 1;
   7, 3, 2, 1;
   9, 4, 2, 1, 1;
  10, 4, 2, 2, 1;
  12, 5, 2, 2, 1, 1;
  15, 6, 3, 2, 1, 1, 1;
  16, 6, 3, 2, 2, 1, 1;
  19, 7, 4, 2, 2, 1, 1, 1;
  21, 8, 4, 2, 2, 2, 1, 1;
  22, 8, 4, 3, 2, 1, 2, 1;
  24, 9, 4, 3, 2, 2, 1, 1, 1;
...
Illustration of initial terms:
Row 1:    _
        _| |
       |_ _|
         2                         Semilength = 2
.
Row 2:      _
           | |
        _ _|_|
       |_ _|1                      Semilength = 3
         2
.
Row 3:          _
               | |
               | |
              _|_|
        _ _ _|                     Semilength = 5
       |_ _ _|2
          3
.
Row 4:              _
                   | |
                   | |
                   | |
                  _|_|
                _|
        _ _ _ _| 1                 Semilength = 7
       |_ _ _ _|2
           4
.
Row 5:                         _
                              | |
                              | |
                              | |
                              | |
                              | |
                           _ _|_|
                         _|
                       _|1         Semilength = 11
                      |1
           _ _ _ _ _ _|
          |_ _ _ _ _ _|3
                6
.
The area (also the number of cells) of the successive diagrams gives A008864.

Crossrefs

Row sums give A000040.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A008864, A018252, A237591, A237593, A245092, A249351, A262626.

A346872 Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 1, 9, 3, 2, 1, 1, 17, 6, 3, 2, 2, 1, 1, 33, 11, 6, 4, 2, 2, 2, 1, 2, 1, 65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 257, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3
Offset: 1

Views

Author

Omar E. Pol, Aug 06 2021

Keywords

Comments

The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.
So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.
Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.
For the definition of "width" see A249351.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).
T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.

Examples

			Triangle begins:
    1;
    2;
    3,  1;
    5,  2,  1;
    9,  3,  2,  1, 1;
   17,  6,  3,  2, 2, 1, 1;
   33, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   65, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:  _
       |_|                              Semilength = 1
        1
Row 2:    _
        _| |
       |_ _|
         2                              Semilength = 2
.
Row 3:        _
             | |
            _| |
        _ _|  _|
       |_ _ _|1                         Semilength = 4
          3
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
                _|  _ _|
               |  _|
        _ _ _ _| | 1                    Semilength = 8
       |_ _ _ _ _|2
            5
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                            _|  _|
                        _ _|  _|        Semilength = 16
                       |  _ _|1 1
                       | | 2
        _ _ _ _ _ _ _ _| |3
       |_ _ _ _ _ _ _ _ _|
                9
.
The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.

Crossrefs

Row sums give A000079.
Column 1 gives A094373.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A000225, A237591, A237593, A245092, A249351, A262626.

A346874 Irregular triangle read by rows in which row n lists the row 2^n - 1 of A237591, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 3, 2, 1, 1, 16, 6, 3, 2, 2, 1, 1, 32, 11, 6, 4, 2, 2, 2, 1, 2, 1, 64, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 256, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3
Offset: 1

Views

Author

Omar E. Pol, Aug 06 2021

Keywords

Comments

The Mersenne number A000225(n) does not has a characteristic shape of its symmetric representation of sigma(A000225(n)). On the other hand, we can find that number in two ways in the symmetric representation of the powers of 2 as follows: the Mersenne numbers are the semilength of the smallest Dyck path and also they equals the area (or the number of cells) of the region of the diagram (see examples).
Therefore we can see a geometric pattern of the distribution of the Mersenne numbers in the stepped pyramid described in A245092.
T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A000225(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000225(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= Mersenne number A000225(n) into k consecutive parts, and the total number of partitions of all positive integers <= Mersenne number A000225(n) into k + 1 consecutive parts.

Examples

			Triangle begins:
    1;
    2,  1;
    4,  2,  1;
    8,  3,  2,  1, 1;
   16,  6,  3,  2, 2, 1, 1;
   32, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   64, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  128, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:
       0_                               Semilength = 0    Area = 1
       |_|
Row 2:
          _
       1_| |                            Semilength = 1    Area = 3
       |_ _|
.
Row 3:        _
             | |
         1  _| |
       2_ _|  _|                        Semilength = 3    Area = 7
       |_ _ _|
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
              1 _|  _ _|
          4   2|  _|                    Semilength = 7    Area = 15
        _ _ _ _| |
       |_ _ _ _ _|
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                         1 1_|  _|
                      2 _ _|  _|        Semilength = 15   Area = 31
                       |  _ _|
               8      3| |
        _ _ _ _ _ _ _ _| |
       |_ _ _ _ _ _ _ _ _|
.

Crossrefs

Row sums give A000225, n >= 1.
Column 1 gives A000079.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
Cf. A000203, A237591, A237593, A245092, A249351, A262626.

A346864 Irregular triangle read by rows in which row n lists the row A014105(n) of A237591, n >= 1.

Original entry on oeis.org

2, 1, 6, 2, 1, 1, 11, 4, 3, 1, 1, 1, 19, 6, 4, 2, 2, 1, 1, 1, 28, 10, 5, 3, 3, 2, 1, 1, 1, 1, 40, 13, 7, 5, 3, 2, 2, 2, 1, 1, 1, 1, 53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1, 69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Omar E. Pol, Aug 17 2021

Keywords

Comments

The characteristic shape of the symmetric representation of sigma(A014105(n)) consists in that in the main diagonal of the diagram the smallest Dyck path has a peak and the largest Dyck path has a valley.
So knowing this characteristic shape we can know if a number is a second hexagonal number (or not) just by looking at the diagram, even ignoring the concept of second hexagonal number.
Therefore we can see a geometric pattern of the distribution of the second hexagonal numbers in the stepped pyramid described in A245092.
T(n,k) is also the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(A014105(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A014105(n).
T(n,k) is also the difference between the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k consecutive parts, and the total number of partitions of all positive integers <= n-th second hexagonal number into exactly k + 1 consecutive parts.
1 together with the first column gives A317186. - Michel Marcus, Jan 12 2025

Examples

			Triangle begins:
   2,  1;
   6,  2,  1, 1;
  11,  4,  3, 1, 1, 1;
  19,  6,  4, 2, 2, 1, 1, 1;
  28, 10,  5, 3, 3, 2, 1, 1, 1, 1;
  40, 13,  7, 5, 3, 2, 2, 2, 1, 1, 1, 1;
  53, 18, 10, 5, 4, 3, 3, 2, 1, 2, 1, 1, 1, 1;
  69, 23, 12, 7, 5, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  86, 29, 15, 9, 6, 5, 4, 2, 3, 2, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
Column h gives the n-th second hexagonal number (A014105).
Column S gives the sum of the divisors of the second hexagonal numbers which equals the area (and the number of cells) of the associated diagram.
--------------------------------------------------------------------------------------
  n   h   S   Diagram
--------------------------------------------------------------------------------------
                  _             _                     _                             _
                 | |           | |                   | |                           | |
              _ _|_|           | |                   | |                           | |
  1   3   4  |_ _|1            | |                   | |                           | |
               2               | |                   | |                           | |
                            _ _| |                   | |                           | |
                           |  _ _|                   | |                           | |
                        _ _|_|                       | |                           | |
                       |  _|1                        | |                           | |
              _ _ _ _ _| | 1                         | |                           | |
  2  10  18  |_ _ _ _ _ _|2                          | |                           | |
                   6                          _ _ _ _|_|                           | |
                                             | |                                   | |
                                            _| |                                   | |
                                           |  _|                                   | |
                                        _ _|_|                                     | |
                                    _ _|  _|1                                      | |
                                   |_ _ _|1 1                                      | |
                                   |  3                               _ _ _ _ _ _ _| |
                                   |4                                |    _ _ _ _ _ _|
              _ _ _ _ _ _ _ _ _ _ _|                                 |   |
  3  21  32  |_ _ _ _ _ _ _ _ _ _ _|                              _ _|   |
                       11                                        |       |
                                                                _|    _ _|
                                                               |     |
                                                            _ _|    _|
                                                        _ _|      _|
                                                       |        _|1
                                                  _ _ _|    _ _|1 1
                                                 |         | 2
                                                 |  _ _ _ _|2
                                                 | |   4
                                                 | |
                                                 | |6
                                                 | |
              _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
  4  36  91  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
                               19
.

Crossrefs

Row sums give A014105, n >= 1.
Row lengths give A005843.
For the characteristic shape of sigma(A000040(n)) see A346871.
For the characteristic shape of sigma(A000079(n)) see A346872.
For the characteristic shape of sigma(A000217(n)) see A346873.
For the visualization of Mersenne numbers A000225 see A346874.
For the characteristic shape of sigma(A000384(n)) see A346875.
For the characteristic shape of sigma(A000396(n)) see A346876.
For the characteristic shape of sigma(A008588(n)) see A224613.
For the characteristic shape of sigma(A174973(n)) see A317305.
Cf. A000203, A237591, A237593, A245092, A249351, A262626, A317186.

Programs

  • PARI
    row(n) = my(m=n*(2*n + 1)); vector((sqrtint(8*m+1)-1)\2, k, ceil((m+1)/k - (k+1)/2) - ceil((m+1)/(k+1) - (k+2)/2)); \\ Michel Marcus, Jan 12 2025

A347155 Sum of divisors of nontriangular numbers.

Original entry on oeis.org

3, 7, 6, 8, 15, 13, 12, 28, 14, 24, 31, 18, 39, 20, 42, 36, 24, 60, 31, 42, 40, 30, 72, 32, 63, 48, 54, 48, 38, 60, 56, 90, 42, 96, 44, 84, 72, 48, 124, 57, 93, 72, 98, 54, 120, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140
Offset: 1

Views

Author

Omar E. Pol, Aug 20 2021

Keywords

Comments

The characteristic shape of the symmetric representation of a(n) consists in that in the main diagonal of the diagram both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys.
So knowing this characteristic shape we can know if a number is a nontriangular number (or not) just by looking at the diagram, even ignoring the concept of nontriangular number.
Therefore we can see a geometric pattern of the distribution of the nontriangular numbers in the stepped pyramid described in A245092.
If both Dyck paths have peaks on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317303.
If both Dyck paths have valleys on the main diagonal then the related subsequence of nontriangular numbers A014132 is A317304.

Examples

			a(6) = 13 because the sum of divisors of the 6th nontriangular (i.e., 9) is 1 + 3 + 9 = 13.
On the other we can see that in the main diagonal of the diagrams both Dyck paths have the same orientation, that is both Dyck paths have peaks or both Dyck paths have valleys as shown below.
Illustration of initial terms:
m(n) = A014132(n).
.
   n   m(n) a(n)   Diagram
.                    _   _ _   _ _ _   _ _ _ _   _ _ _ _ _   _ _ _ _ _ _
                   _| | | | | | | | | | | | | | | | | | | | | | | | | | |
   1    2    3    |_ _|_| | | | | | | | | | | | | | | | | | | | | | | | |
                   _ _|  _|_| | | | | | | | | | | | | | | | | | | | | | |
   2    4    7    |_ _ _|    _|_| | | | | | | | | | | | | | | | | | | | |
   3    5    6    |_ _ _|  _|  _ _|_| | | | | | | | | | | | | | | | | | |
                   _ _ _ _|  _| |  _ _|_| | | | | | | | | | | | | | | | |
   4    7    8    |_ _ _ _| |_ _|_|    _ _|_| | | | | | | | | | | | | | |
   5    8   15    |_ _ _ _ _|  _|     |  _ _ _|_| | | | | | | | | | | | |
   6    9   13    |_ _ _ _ _| |      _|_| |  _ _ _|_| | | | | | | | | | |
                   _ _ _ _ _ _|  _ _|    _| |    _ _ _|_| | | | | | | | |
   7   11   12    |_ _ _ _ _ _| |  _|  _|  _|   |  _ _ _ _|_| | | | | | |
   8   12   28    |_ _ _ _ _ _ _| |_ _|  _|  _ _| | |  _ _ _ _|_| | | | |
   9   13   14    |_ _ _ _ _ _ _| |  _ _|  _|    _| | |    _ _ _ _|_| | |
  10   14   24    |_ _ _ _ _ _ _ _| |     |     |  _|_|   |  _ _ _ _ _|_|
                   _ _ _ _ _ _ _ _| |  _ _|  _ _|_|       | | |
  11   16   31    |_ _ _ _ _ _ _ _ _| |  _ _|  _|      _ _|_| |
  12   17   18    |_ _ _ _ _ _ _ _ _| | |_ _ _|      _| |  _ _|
  13   18   39    |_ _ _ _ _ _ _ _ _ _| |  _ _|    _|  _|_|
  14   19   20    |_ _ _ _ _ _ _ _ _ _| | |       |_ _|
  15   20   42    |_ _ _ _ _ _ _ _ _ _ _| |  _ _ _|  _|
                   _ _ _ _ _ _ _ _ _ _ _| | |  _ _| |
  16   22   36    |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _|
  17   23   24    |_ _ _ _ _ _ _ _ _ _ _ _| | |
  18   24   60    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  19   25   31    |_ _ _ _ _ _ _ _ _ _ _ _ _| |
  20   26   42    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
  21   27   40    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Column m gives the nontriangular numbers.
Also the diagrams have on the main diagonal the following property: diagram [1] has peaks, diagrams [2, 3] have valleys, diagrams [4, 5, 6] have peaks, diagrams [7, 8, 9, 10] have valleys, and so on.
a(n) is also the area (and the number of cells) of the n-th diagram.
For n = 3 the sum of the regions (or parts) of the third diagram is 3 + 3 = 6, so a(3) = 6.
For more information see A237593.

Crossrefs

Cf. A000203, A000217, A014132, A237591, A237593, A245092, A262626, A317303, A317304, A346873.
Some sequences that gives sum of divisors: A000225 (of powers of 2), A008864 (of prime numbers), A065764 (of squares), A073255 (of composites), A074285 (of triangular numbers, also of generalized hexagonal numbers), A139256 (of perfect numbers), A175926 (of cubes), A224613 (of multiples of 6), A346865 (of hexagonal numbers), A346866 (of second hexagonal numbers), A346867 (of numbers with middle divisors), A346868 (of numbers with no middle divisors).

Programs

Formula

a(n) = A000203(A014132(n)).
Showing 1-8 of 8 results.

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