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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.
7
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
1,4
COMMENTS
For further information about the correspondence divisor/part see A338156.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)
EXAMPLE
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.
MATHEMATICA
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 *)
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 26 2020
STATUS
approved