A127093 -id:A127093 - cp's OEIS frontend

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

1, 1, 2, 1, 1, 0, 3, 1, 2, 1, 1, 1, 2, 0, 4, 1, 0, 3, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A127094 Triangle, reversal of A127093.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 4, 0, 2, 1, 5, 0, 0, 0, 1, 6, 0, 0, 3, 2, 1, 7, 0, 0, 0, 0, 0, 1, 8, 0, 0, 0, 4, 0, 2, 1, 9, 0, 0, 0, 0, 0, 3, 0, 1, 10, 0, 0, 0, 0, 5, 0, 0, 2, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 05 2007

			First few rows of the triangle are:
  1;
  2, 1;
  3, 0, 1;
  4, 0, 2, 1;
  5, 0, 0, 0, 1;
  6, 0, 0, 3, 2, 1;
  ...

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A000203 (row sums), A126988, A127093.

[n mod (k-n-1) - (n+1) mod (k-n-1) + 1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 08 2021

Table[Mod[n, k-n-1] - Mod[n+1, k-n-1] +1, {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 08 2021 *)

flatten([[n%(k-n-1) - (n+1)%(k-n-1) + 1 for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Mar 08 2021

Reversed rows of A127093.

T(n, K) = mod(n, k-n-1) - mod(n+1, k-n-1) + 1. - Mats Granvik, Sep 02 2007

A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 2, 1, 2, 0, 4, 1, 0, 3, 2, 4, 3, 1, 0, 0, 0, 5, 1, 2, 0, 4, 2, 0, 6, 3, 6, 5, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 2, 4, 0, 8, 3, 0, 9, 5, 10, 7, 1, 0, 0, 0, 0, 0, 7, 1, 2, 3, 0, 0, 6, 2, 0, 0, 0, 10, 3, 6, 0, 12, 5, 0, 15, 7, 14, 11, 1, 2, 0, 4, 0, 0, 0, 8

Offset: 1

Omar E. Pol, Dec 26 2020

This triangle is a condensed version of the more irregular triangle A340031.

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
[1];
[1, 2],          [1];
[1, 0, 3],       [1, 2],       [2];
[1, 2, 0, 4],    [1, 0, 3],    [2, 4],    [3];
[1, 0, 0, 0, 5], [1, 2, 0, 4], [2, 0, 6], [3, 6], [5];
[...
Row sums give A066186.
Written as an irregular tetrahedron the first five slices are:
--
1;
-----
1, 2,
1;
--------
1, 0, 3,
1, 2,
2;
-----------
1, 2, 0, 4,
1, 0, 3,
2, 4,
3;
--------------
1, 0, 0, 0, 5,
1, 2, 0, 4,
2, 0, 6,
3, 6,
5;
--------------
Row sums give A339106.
The following table formed by four zones shows the correspondence between divisor and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
| C | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
| O |    -    |     |       |  2      |  2 4      |  2 0 6      |
| N |    -    |     |       |         |  3        |  3 6        |
| D |    -    |     |       |         |           |  5          |
|---|---------|-----|-------|---------|-----------|-------------|
.
This lower zone of the table is a condensed version of the "divisors" zone.

Paolo Xausa, Table of n, a(n) for n = 1..11480 (rows 1..40 of the triangle, flattened)

Row sums give A066186.
Nonzero terms give A340056.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340011row[n_]:=Flatten[Table[A127093row[n-m+1]PartitionsP[m-1],{m,n}]];
Array[A340011row,10] (* Paolo Xausa, Sep 28 2023 *)

A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 3, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 6

Offset: 1

Omar E. Pol, Dec 26 2020

For further information about the correspondence divisor/part see A338156.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 1, 2, 1, 0, 3;
  1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 2, 0, 4;
  1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 3, 1, 0, 3, 1, 2, 0, 4, 1, 0, 0, 0, 5;
  ...
Written as an irregular tetrahedron the first five slices are:
  1;
  --
  1,
  1, 2;
  -----
  1,
  1,
  1, 2,
  1, 0, 3;
  --------
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 2, 0, 4;
  -----------
  1,
  1,
  1,
  1,
  1,
  1, 2,
  1, 2,
  1, 2,
  1, 0, 3,
  1, 0, 3,
  1, 2, 0, 4,
  1, 0, 0, 0, 5;
  --------------
  ...
The slices of the tetrahedron appear in the upper zone of the following table (formed by three zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
| D | A127093 |     |       |         |           |  1          |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A127093 |     |       |         |  1        |  1 2        |
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| S | A127093 |     |       |  1      |  1 2      |  1 0 3      |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
| L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
| I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
| A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| R |         |     |       |  3      |  3 1      |  3 1 1      |
| T |         |     |       |         |  2 2      |  2 2 1      |
| I |         |     |       |         |  4        |  4 1        |
| T |         |     |       |         |           |  3 2        |
| I |         |     |       |         |           |  5          |
| O |         |     |       |         |           |             |
| N |         |     |       |         |           |             |
| S |         |     |       |         |           |             |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A340035 but here, in the upper zone, every row is A127093 instead of A027750.
Also the above table is the table of A340031 upside down.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A340035.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A338156, A339106, A339258, A339278, A339304, A340031, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340032row[n_]:=Flatten[Table[ConstantArray[A127093row[m],PartitionsP[n-m]],{m,n}]];
Array[A340032row,7] (* Paolo Xausa, Sep 28 2023 *)

A127098 Triangle T(n,m) read by rows: product A127093 * A127094.

Original entry on oeis.org

1, 5, 2, 10, 0, 3, 21, 2, 8, 4, 26, 0, 0, 0, 5, 50, 2, 3, 18, 12, 6, 50, 0, 0, 0, 0, 0, 7, 85, 2, 8, 4, 32, 0, 16, 8, 91, 0, 3, 0, 0, 0, 27, 0, 9, 130, 2, 0, 0, 5, 50, 0, 0, 20, 10, 122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 210, 2, 11, 22, 12, 6, 72, 0, 48, 36, 24, 12, 170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 250, 2, 0, 0, 0, 0, 7, 98, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

A127097 is a close companion.

			First few rows of the triangle are:
1;
5, 2;
10, 0, 3;
21, 2, 8, 4;
26, 0, 0, 0, 5;
50, 2, 3, 18, 12, 6;
50, 0, 0, 0, 0, 0, 7;
85, 2, 8, 4, 32, 0, 16, 8;

Cf. A127097, A127093, A127094, A001001, A001157.

A127093 := proc(n,m) if m> n or m<1 or n < 1 then 0 ; elif (n mod m) = 0 then m; else 0 ; fi; end:
A127094 := proc(n,m) A127093(n,n-m+1) ; end: A127098 := proc(n,m) add( A127093(n,k)*A127094(k,m),k=1..n) ; end:
for n from 1 to 30 do for m from 1 to n do printf("%d,",A127098(n,m)) ; od: od: # R. J. Mathar, Mar 02 2009

T(n,m) = Sum_{k=m..n} A127093(n,k)*A127094(k,m).
Row sums: Sum_{m=1..n} T(n,m) = A001001(n).
Left column: T(1,m) = A001157(m).

Edited and extended by R. J. Mathar, Mar 02 2009

A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

Original entry on oeis.org

1, 5, 2, 10, 0, 3, 21, 10, 0, 4, 26, 0, 0, 0, 5, 50, 20, 15, 0, 0, 6, 50, 0, 0, 0, 0, 0, 7, 85, 42, 0, 20, 0, 0, 0, 8, 91, 0, 30, 0, 0, 0, 0, 0, 9, 130, 52, 0, 0, 25, 0, 0, 0, 0, 10, 122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 210, 100, 63, 40, 0, 30, 0, 0, 0, 0, 0, 12, 170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A127093 and A126988.

			First few rows of the triangle are:
1;
5, 2;
10, 0, 3;
21, 10, 0, 4;
26, 0, 0, 0, 5;
50, 20, 15, 0, 0, 6;
50, 0, 0, 0, 0, 0, 7;
...

Cf. A126988, A000203, A127098, A001187, A001001 (row sums), A127093.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A126988 := proc(n,k) if n mod k = 0 then n/k; else 0; fi; end:
A127097 := proc(n,m) add( A127093(n,j)*A126988(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127097(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T(n,m) = sum_{j=m..n} A127093(n,j)*A126988(j,m).

T(n,1) = A001157(n).

A-numbers corrected by R. J. Mathar, Aug 18 2009

A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

Original entry on oeis.org

1, 3, 2, 4, 0, 3, 7, 6, 0, 4, 6, 0, 0, 0, 5, 12, 8, 9, 0, 0, 6, 8, 0, 0, 0, 0, 0, 7, 15, 14, 0, 12, 0, 0, 0, 8, 13, 0, 12, 0, 0, 0, 0, 0, 9, 18, 12, 0, 0, 15, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 28, 24, 21, 16, 0, 18, 0, 0, 0, 0, 0, 12, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 24, 16, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A126988 and A127093.

			First few rows of the triangle are:
1;
3, 2;
4, 0, 3;
7, 6, 0, 4;
6, 0, 0, 0, 5;
12, 8, 9, 0, 0, 6;
8, 0, 0, 0, 0, 0, 7;
15, 14, 0, 12, 0, 0, 0, 8;
13, 0, 12, 0, 0, 0, 0, 0, 9;
18, 12, 0, 0, 15, 0, 0, 0, 0, 10;
...

Cf. A060640 (row sums), A000203, A127093, A127094, A123229, A127096, A127097, A127098, A127013, A127057.

T(n,m) = sum_{j=m..n} A126988(n,j)*A127093(j,m).

T(n,1) = A000203(n).

Extended by R. J. Mathar, Aug 18 2009

A130055 A129691 * A127093.

Original entry on oeis.org

1, 0, 2, -1, 0, 3, -1, 0, 0, 4, -3, 0, 0, 0, 5, 0, -2, 0, 0, 0, 6, -5, 0, 0, 0, 0, 0, 7, -2, -2, 0, 0, 0, 0, 0, 8, -3, 0, -3, 0, 0, 0, 0, 0, 9, 0, -6, 0, 0, 0, 0, 0, 0, 0, 10

Offset: 1

Gary W. Adamson, May 04 2007

Row sums = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...). Left column = A130054: (1, 0, -1, -1, -3, 0, -5, -2, -3, 0, ...).

			First few rows of the triangle:
   1;
   0,  2;
  -1,  0,  3;
  -1,  0,  0,  4;
  -3,  0,  0,  0,  5;
   0, -2,  0,  0,  0,  6;
  -5,  0,  0,  0,  0,  0,  7;
  ...

Cf. A129691, A127093, A000005, A130054.

A129691 * A127093 as infinite lower triangular matrices.

A134559 A127093 * A000012.

Original entry on oeis.org

1, 3, 2, 4, 3, 3, 7, 6, 4, 4, 6, 5, 5, 5, 5, 12, 11, 9, 6, 6, 6, 8, 7, 7, 7, 7, 7, 7, 15, 14, 12, 12, 8, 8, 8, 8, 13, 12, 12, 9, 9, 9, 9, 9, 9, 18, 17, 15, 15, 15, 10, 10, 10, 10, 10

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = A001157: (1, 5, 10, 21, 26, 50, ...). Left border = sigma(n), A000203.
From Lechoslaw Ratajczak, Nov 01 2019: (Start)
Let b_n(k) (n = 1,2,3,...) be consecutive finite sequences defined as follows: b_n(k) is the sum of all integers u satisfying the equation: n mod u = k (1 <= u <= n, k = 0,1,2,...,ceiling(n/2)-1). These sequences are consecutive antidiagonals of the triangle (b_n(k) = T(n-k,k+1)).
The example for n = 8 (k_max = ceiling(8/2) - 1 = 3):
- b_8(0) = T(8-0,0+1) = T(8,1) = 15 = sigma(8) because 8 mod {1,2,4,8} = 0 and 1+2+4+8 = 15;
- b_8(1) = T(8-1,1+1) = T(7,2) = 7 = A039653(8-1) because 8 mod 7 = 1;
- b_8(2) = T(8-2,2+1) = T(6,3) = 9 because 8 mod {3,6} = 2 and 3+6 = 9;
- b_8(3) = T(8-3,3+1) = T(5,4) = 5 because 8 mod 5 = 3.
Conjecture: let P(n) be the n-th antidiagonal product (P(n) = Product_{k=0..ceiling(n/2)-1} b_n(k)). Consecutive n satisfying two equations: gcd(P(n),n) = 1 and gcd(P(n+1),n+1) = 2 are consecutive elements of A005383 (primes p such that (p+1)/2 are also primes, save A005383(1) = 3 and A005383(2) = 5). The conjecture is false if for any prime number p belonging to A005383 gcd(P(p),p) = p. The conjecture was checked for 2000 consecutive integers. (End)

			First few rows of the triangle are:
   1;
   3,  2;
   4,  3,  3;
   7,  6,  4,  4;
   6,  5,  5,  5, 5;
  12, 11,  9,  6, 6, 6;
   8,  7,  7,  7, 7, 7, 7;
  15, 14, 12, 12, 8, 8, 8, 8;
  ...

Cf. A127093, A001157, A000203.

A127093 * A000012 as infinite lower triangular matrices. Triangle read by rows, partial sums of A127093 terms starting from the right.

A253951 A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731A127093Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).

Original entry on oeis.org

0, 1, 5, 9, 20, 23, 42, 52, 69, 77, 113, 119, 165, 177, 190, 214, 279, 291, 366, 379, 399, 422, 517, 533, 599, 625, 679, 701, 829, 846, 986, 1035, 1069, 1105, 1137, 1164, 1339, 1380, 1417, 1449, 1646, 1674, 1883, 1918, 1955, 2008, 2239, 2274, 2420, 2462, 2515, 2559, 2827, 2874, 2929

Offset: 1

Mats Granvik, Jan 20 2015

nonn

a(n) ~ log(A003418(n))*n, based on the comment by Hans Havermann in A048272 referring to an argument by Gareth McCaughan.
The exact relation is: lim_{n->Infinity} log(A003418(k))*n = Sum_{x=1..n} Sum_{y=1..k} T(x,y), where T is the matrix product: T = A051731*A127093*Transpose(A054524) and T(n,1)=0.
Compare a(n) to round(log(A003418)*n)= 0, 1, 5, 10, 20, 25, 42, 54, 70, 78,...

Robert G. Wilson v, Table of n, a(n) for n = 1..1002

with(LinearAlgebra):
N:= 200:
A051731:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, 1, 0),shape=triangular[lower]):
A127093:= Matrix(N,N,(n,k) -> `if`(n mod k = 0, k, 0), shape=triangular[lower]):
A054524T:= Matrix(N,N,(k,n) -> `if`(n mod k = 0, numtheory:-mobius(k),0), shape=triangular[upper]):
T:= A051731 . A127093 . A054524T:
a[1]:= 0:
for n from 2 to N do
  a[n]:= a[n-1] + add(T[i,n],i=1..n) + add(T[n,j],j=2..n-1)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 20 2015

nn = 55;
Z = Table[ If[ Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
A = Table[ If[ Mod[n, k] == 0, k, 0], {n, nn}, {k, nn}];
B = Table[ If[ Mod[n, k] == 0, MoebiusMu[k], 0], {n, nn}, {k, nn}];
MatrixForm[T = Z.A.Transpose[B]];
T[[All, 1]] = 0;
a = Table[ Total[ T[[1 ;; n, 1 ;; n]], 2], {n, nn}]
(* shows a graph *) Show[ ListLinePlot[a], ListLinePlot[ Accumulate[ MangoldtLambda[ Range[ nn]]]]]

a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T=A051731*A127093*Transpose(A054524) and T(n,1)=0. (* stands for matrix multiplication)

cp's OEIS Frontend

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

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A127094 Triangle, reversal of A127093.

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A340011 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the j-th row of triangle A127093 but with every term multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.

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A340032 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(n-m) copies of the row m of triangle A127093, with 1 <= m <= n.

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A127098 Triangle T(n,m) read by rows: product A127093 * A127094.

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A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

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Maple

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A127099 Triangle T(n,m) = A126988 *A127093 read by rows.

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A130055 A129691 * A127093.

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A253951 A partial double sum of integers: a(n) = Sum_{x=1..n} Sum_{y=1..n} T(x,y), where T is the matrix product: T = A051731A127093Transpose(A054524) and T(n,1)=0 (* stands for matrix multiplication).