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A340056
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Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of the divisors of j multiplied by A000041(m-1), where j = n - m + 1 and 1 <= m <= n.
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7
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1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 4, 1, 3, 2, 4, 3, 1, 5, 1, 2, 4, 2, 6, 3, 6, 5, 1, 2, 3, 6, 1, 5, 2, 4, 8, 3, 9, 5, 10, 7, 1, 7, 1, 2, 3, 6, 2, 10, 3, 6, 12, 5, 15, 7, 14, 11, 1, 2, 4, 8, 1, 7, 2, 4, 6, 12, 3, 15, 5, 10, 20, 7, 21, 11, 22, 15, 1, 3, 9, 1, 2, 4, 8, 2, 14, 3, 6, 9, 18, 5
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OFFSET
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1,3
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COMMENTS
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This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.
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LINKS
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EXAMPLE
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Triangle begins:
[1];
[1, 2], [1];
[1, 3], [1, 2], [2];
[1, 2, 4], [1, 3], [2, 4], [3];
[1, 5], [1, 2, 4], [2, 6], [3, 6], [5];
[...
The row sums of triangle give A066186.
Written as an irregular tetrahedron the first five slices are:
1;
-----
1, 2,
1;
-----
1, 3,
1, 2,
2;
--------
1, 2, 4,
1, 3,
2, 4,
3;
--------
1, 5,
1, 2, 4,
2, 6,
3, 6,
5;
--------
The row sums of tetrahedron give A339106.
The slices of the tetrahedron appear in 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 |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
| |---------|-----|-------|---------|-----------|-------------|
| | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| |---------|-----|-------|---------|-----------|-------------|
| D | A027750 | | | 1 | 1 2 | 1 3 |
| I | A027750 | | | 1 | 1 2 | 1 3 |
| V |---------|-----|-------|---------|-----------|-------------|
| R |---------|-----|-------|---------|-----------|-------------|
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
| C | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
| O | - | | | 2 | 2 4 | 2 6 |
| N | - | | | | 3 | 3 6 |
| D | - | | | | | 5 |
|---|---------|-----|-------|---------|-----------|-------------|
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The lower zone is a condensed version of the "divisors" zone.
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MATHEMATICA
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A340056row[n_]:=Flatten[Table[Divisors[n-m]PartitionsP[m], {m, 0, n-1}]]; Array[A340056row, 10] (* Paolo Xausa, Sep 01 2023 *)
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CROSSREFS
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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, A336812, A337209, A338156, A339106, A339258, A339278, A339304, 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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