A346533 -id:A346533 - cp's OEIS frontend

A092403 a(n) = sigma(n) + sigma(n+1).

4, 7, 11, 13, 18, 20, 23, 28, 31, 30, 40, 42, 38, 48, 55, 49, 57, 59, 62, 74, 68, 60, 84, 91, 73, 82, 96, 86, 102, 104, 95, 111, 102, 102, 139, 129, 98, 116, 146, 132, 138, 140, 128, 162, 150, 120, 172, 181, 150, 165, 170, 152, 174, 192, 192, 200, 170, 150, 228, 230

Offset: 1

Jon Perry, Mar 22 2004

nonn

a(n) is the area of the terrace in the level 1 of the polycube called "tower" described in A221529 where the longest side of its base is equal to n + 1, thus a(n) has a symmetric representation. - Omar E. Pol, Jul 22 2021

Cf. A000203, A092405, A221529, A237593, A346533.

Total/@Partition[DivisorSigma[1,Range[70]],2,1] (* Harvey P. Dale, Feb 19 2018 *)

for(i=1,60,print1(","sigma(i)+sigma(i+1)))

a(n) = A346533(n+1,n). - Omar E. Pol, Jul 22 2021

A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

Original entry on oeis.org

1, 4, 8, 1, 15, 4, 1, 21, 8, 4, 1, 1, 33, 15, 8, 4, 4, 1, 1, 41, 21, 15, 8, 8, 4, 4, 1, 1, 1, 1, 56, 33, 21, 15, 15, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 87, 56, 41, 33, 33, 21, 21, 15, 15, 15, 15, 8, 8, 8, 8

Offset: 1

Omar E. Pol, Jan 07 2021

T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the tower described in A221529 whose largest side of the base is equal to n (see example). - Omar E. Pol, Jan 08 2022

			Triangle begins:
   1;
   4;
   8,  1;
  15,  4,  1;
  21,  8,  4,  1,  1;
  33, 15,  8,  4,  4,  1,  1;
  41, 21, 15,  8,  8,  4,  4, 1, 1, 1, 1;
  56, 33, 21, 15, 15,  8,  8, 4, 4, 4, 4, 1, 1, 1, 1;
  69, 41, 33, 21, 21, 15, 15, 8, 8, 8, 8, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
...
For n = 9 the length of row 9 is A000041(9-1) = 22.
From _Omar E. Pol_, Jan 08 2022: (Start)
For n = 9 the lateral view and top view of the tower described in A221529 look like as shown below:
                        _
    22        1        | |
    21        1        | |
    20        1        | |
    19        1        | |
    18        1        | |
    17        1        | |
    16        1        |_|_
    15        4        |   |
    14        4        |   |
    13        4        |   |
    12        4        |_ _|_
    11        8        |   | |
    10        8        |   | |
     9        8        |   | |
     8        8        |_ _|_|_
     7       15        |     | |
     6       15        |_ _ _| |_
     5       21        |     |   |
     4       21        |_ _ _|_ _|_
     3       33        |_ _ _ _| | |_
     2       41        |_ _ _ _|_|_ _|_ _
     1       69        |_ _ _ _ _|_ _|_ _|
.
   Level   Row 9         Lateral view
     k     T(9,k)        of the tower
.
                        _ _ _ _ _ _ _ _ _
                       |_| | | | | | |   |
                       |_ _|_| | | | |   |
                       |_ _|  _|_| | |   |
                       |_ _ _|    _|_|   |
                       |_ _ _|  _|    _ _|
                       |_ _ _ _|     |
                       |_ _ _ _|  _ _|
                       |         |
                       |_ _ _ _ _|
.
                           Top view
                         of the tower
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the tower, so T(9,1) = 69.
For n = 9 and k = 22 there is only one cubic cell in the level 22 (the top) of the tower, so T(9,22) = 1.
The volume of the tower (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000041 hence it's equal to A066186(9) = 270, equaling the sum of the 9th row of triangle. (End)

Row sums give A066186.
Row lengths give A000041.
The length of the m-th block in row n is A187219(m), m >= 1.
Cf. A350637 (analog for the stepped pyramid described in A245092).
Cf. A000203, A024916, A196020, A221529, A236104, A235791, A237270, A237271, A237593, A339278, A262626, A336811, A338156, A340035, A341149, A346533, A350333.

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--; ); 1+s; } \\ A336811
g(n) = sum(k=1, n, n\k*k); \\ A024916
row(n) = vector(f(n), k, g(T(n,k))); \\ Michel Marcus, Jan 22 2022

T(n,k) = A024916(A336811(n,k)).

T(n,k) = Sum_{j=1..n} A339278(j,k). - Omar E. Pol, Jan 08 2022

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Original entry on oeis.org

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1

Offset: 1

Omar E. Pol, Jan 12 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

Original entry on oeis.org

1, 3, 1, 4, 1, 2, 5, 1, 2, 2, 5, 1, 2, 2, 3, 6, 1, 2, 2, 3, 2, 6, 1, 2, 2, 3, 2, 4, 6, 1, 2, 2, 3, 2, 4, 2, 7, 1, 2, 2, 3, 2, 4, 2, 4, 7, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6

Offset: 1

Omar E. Pol, Jul 23 2021

T(n,k) is the total number of divisors related to the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower described in A221529.

			Triangle begins:
1;
3;
1, 4;
1, 2, 5;
1, 2, 2, 5;
1, 2, 2, 3, 6;
1, 2, 2, 3, 2, 6;
1, 2, 2, 3, 2, 4, 6;
1, 2, 2, 3, 2, 4, 2, 7;
1, 2, 2, 3, 2, 4, 2, 4, 7;
1, 2, 2, 3, 2, 4, 2, 4, 3, 6;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 8;
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 6;
...

The length of row n is A028310(n-1).
Row sums give A006218, n >= 1.
Leading diagonal gives A092405.
Other diagonals give A000005.
Column 1 gives the absolute values of A260196.
Companion of A346533.
Cf. A221529, A237270, A237593, A336811, A338156, A340035.

cp's OEIS Frontend

A092403 a(n) = sigma(n) + sigma(n+1).

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A340423 Irregular triangle read by rows T(n,k) in which row n has length A000041(n-1) and every column k is A024916, n >= 1, k >= 1.

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A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

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A346562 Irregular triangle read by rows in which row n lists the first n - 2 terms of A000005 together with the sum of A000005(n-1) and A000005(n), with a(1) = 1.

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