OFFSET
1,3
COMMENTS
Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)
EXAMPLE
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.
.
MATHEMATICA
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 *)
CROSSREFS
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.
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 26 2020
STATUS
approved