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A340057
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Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the block m consists of the divisors of m multiplied by A000041(n-m), with 1 <= m <= n.
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5
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1, 1, 1, 2, 2, 1, 2, 1, 3, 3, 2, 4, 1, 3, 1, 2, 4, 5, 3, 6, 2, 6, 1, 2, 4, 1, 5, 7, 5, 10, 3, 9, 2, 4, 8, 1, 5, 1, 2, 3, 6, 11, 7, 14, 5, 15, 3, 6, 12, 2, 10, 1, 2, 3, 6, 1, 7, 15, 11, 22, 7, 21, 5, 10, 20, 3, 15, 2, 4, 6, 12, 1, 7, 1, 2, 4, 8, 22, 15, 30, 11, 33, 7, 14, 28, 5, 25
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OFFSET
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1,4
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COMMENTS
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This triangle is a condensed version of the more irregular triangle A340035.
For further information about the correspondence divisor/part see A338156.
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LINKS
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EXAMPLE
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Triangle begins:
[1];
[1], [1, 2];
[2], [1, 2], [1, 3];
[3], [2, 4], [1, 3], [1, 2, 4];
[5], [3, 6], [2, 6], [1, 2, 4], [1, 5];
[7], [5, 10], [3, 9], [2, 4, 8], [1, 5], [1, 2, 3, 6];
[11], [7, 14], [5, 15], [3, 6, 12], [2, 10], [1, 2, 3, 6], [1, 7];
...
Written as a tetrahedrons the first five slices are:
--
1;
--
1,
1, 2;
-----
2,
1, 2,
1, 3;
-----
3,
2, 4,
1, 3,
1, 2, 4;
--------
5,
3, 6,
2, 6,
1, 2, 4,
1, 5;
--------
The slices of the tetrahedron appear in the upper zone of the following table (formed by four zones) which shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n | | 1 | 2 | 3 | 4 | 5 |
|---|---------|-----|-------|---------|-----------|-------------|
| | - | | | | | 5 |
| C | - | | | | 3 | 3 6 |
| O | - | | | 2 | 2 4 | 2 6 |
| N | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| D | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| I |---------|-----|-------|---------|-----------|-------------|
| 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 |
|---|---------|-----|-------|---------|-----------|-------------|
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|---|---------|-----|-------|---------|-----------|-------------|
| | 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 | | | | | | |
|---|---------|-----|-------|---------|-----------|-------------|
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The upper zone is a condensed version of the "divisors" zone.
The above table is the table of A340056 upside down.
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MATHEMATICA
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A340057row[n_]:=Flatten[Table[Divisors[m]PartitionsP[n-m], {m, n}]]; Array[A340057row, 10] (* Paolo Xausa, Sep 02 2023 *)
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CROSSREFS
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Cf. A000041, A002260, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A221529, A221530, A221531, A236104, A237593, A245092, A245095, A221650, A302246, A302247, A336811, A336812, A337209, A338156, A339106, A339258, A339278, A339304, A340011, A340031, A340032, A340056, A340057, A340061.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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