A340035 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 divisors of m, with 1 <= m <= n.
1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 4, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 3, 1, 2, 4, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2
Offset: 1
Examples
Triangle begins:
1;
1, 1, 2;
1, 1, 1, 2, 1, 3;
1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 4;
1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 5;
...
Written as an irregular tetrahedron the first five slices are:
1;
--
1,
1, 2;
-----
1,
1,
1, 2
1, 3;
-----
1,
1,
1,
1, 2,
1, 2,
1, 3,
1, 2, 4;
--------
1,
1,
1,
1,
1,
1, 2,
1, 2,
1, 2,
1, 3,
1, 3,
1, 2, 4,
1, 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 |
|---|---------|-----|-------|---------|-----------|-------------|
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| | A027750 | | | | | 1 |
| D | A027750 | | | | | 1 |
| I |---------|-----|-------|---------|-----------|-------------|
| V | A027750 | | | | 1 | 1 2 |
| I | A027750 | | | | 1 | 1 2 |
| S | A027750 | | | | 1 | 1 2 |
| O |---------|-----|-------|---------|-----------|-------------|
| R | A027750 | | | 1 | 1 2 | 1 3 |
| S | A027750 | | | 1 | 1 2 | 1 3 |
| |---------|-----|-------|---------|-----------|-------------|
| | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| |---------|-----|-------|---------|-----------|-------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 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 A340032 but here, in the upper zone, every row is A027750 instead of A127093.
Also the above table is the table of A338156 upside down.
The connection with the tower described in A221529 is as follows (n = 7):
|--------|------------------------|
| Level | |
| in the | 7th slice of divisors |
| tower | |
|--------|------------------------|
| 11 | 1, |
| 10 | 1, |
| 9 | 1, |
| 8 | 1, |
| 7 | 1, |
| 6 | 1, |
| 5 | 1, |
| 4 | 1, |
| 3 | 1, |
| 2 | 1, |
| 1 | 1, |
|--------|------------------------|
| 7 | 1, 2, |
| 6 | 1, 2, |
| 5 | 1, 2, |
| 4 | 1, 2, |
| 3 | 1, 2, |
| 2 | 1, 2, |
| 1 | 1, 2, |
|--------|------------------------|
| 5 | 1, 3, |
| 4 | 1, 3, |
| 3 | 1, 3, |
| 2 | 1, 3, | Level
| 1 | 1, 3, | _
|--------|------------------------| 11 | |
| 3 | 1, 2, 4, | 10 | |
| 2 | 1, 2, 4, | 9 | |
| 1 | 1, 2, 4, | 8 |_|_
|--------|------------------------| 7 | |
| 2 | 1, 5, | 6 |_ _|_
| 1 | 1, 5, | 5 | | |
|--------|------------------------| 4 |_ _|_|_
| 1 | 1, 2, 3, 6, | 3 |_ _ _| |_
|--------|------------------------| 2 |_ _ _|_ _|_ _
| 1 | 1, 7; | 1 |_ _ _ _|_|_ _|
|--------|------------------------|
Figure 1. Figure 2.
Lateral view
of the tower.
.
_ _ _ _ _ _ _
|_| | | | | |
|_ _|_| | | |
|_ _| _|_| |
|_ _ _| _ _|
|_ _ _| _|
| |
|_ _ _ _|
.
Figure 3.
Top view
of the tower.
.
Figure 1 shows the terms of the 7th row of the triangle arranged as the 7th slice of the tetrahedron. The left hand column (see figure 1) gives the level of the sum of the divisors in the tower (see figures 2 and 3).
Links
- Paolo Xausa, Table of n, a(n) for n = 1..17815 (rows 1..20 of triangle, flattened)
Crossrefs
Nonzero terms of A340032.
Row lengths give A006128, n >= 1.
Row sums give A066186, n >= 1.
Cf. A000041, A002260, A027750, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A206437, A207031, A207383, A221529, A221530, A221531, A221649, A236104, A237593, A245092, A245095, A221649, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.
Programs
-
Mathematica
A340035row[n_]:=Flatten[Array[ConstantArray[Divisors[#],PartitionsP[n-#]]&,n]]; nrows=7;Array[A340035row,nrows] (* Paolo Xausa, Jun 20 2022 *)
Comments