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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.

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%I A340035 #62 Jun 22 2022 20:34:23
%S A340035 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,
%T A340035 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,
%U A340035 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
%N 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.
%C A340035 For further information about the correspondence divisor/part see A338156.
%H A340035 Paolo Xausa, <a href="/A340035/b340035.txt">Table of n, a(n) for n = 1..17815</a> (rows 1..20 of triangle, flattened)
%e A340035 Triangle begins:
%e A340035   1;
%e A340035   1, 1, 2;
%e A340035   1, 1, 1, 2, 1, 3;
%e A340035   1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 4;
%e A340035   1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 4, 1, 5;
%e A340035   ...
%e A340035 Written as an irregular tetrahedron the first five slices are:
%e A340035   1;
%e A340035   --
%e A340035   1,
%e A340035   1, 2;
%e A340035   -----
%e A340035   1,
%e A340035   1,
%e A340035   1, 2
%e A340035   1, 3;
%e A340035   -----
%e A340035   1,
%e A340035   1,
%e A340035   1,
%e A340035   1, 2,
%e A340035   1, 2,
%e A340035   1, 3,
%e A340035   1, 2, 4;
%e A340035   --------
%e A340035   1,
%e A340035   1,
%e A340035   1,
%e A340035   1,
%e A340035   1,
%e A340035   1, 2,
%e A340035   1, 2,
%e A340035   1, 2,
%e A340035   1, 3,
%e A340035   1, 3,
%e A340035   1, 2, 4,
%e A340035   1, 5;
%e A340035 --------
%e A340035 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):
%e A340035 .
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 | n |         |  1  |   2   |    3    |     4     |      5      |
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 |   | A027750 |     |       |         |           |  1          |
%e A340035 |   | A027750 |     |       |         |           |  1          |
%e A340035 |   | A027750 |     |       |         |           |  1          |
%e A340035 |   | A027750 |     |       |         |           |  1          |
%e A340035 | D | A027750 |     |       |         |           |  1          |
%e A340035 | I |---------|-----|-------|---------|-----------|-------------|
%e A340035 | V | A027750 |     |       |         |  1        |  1 2        |
%e A340035 | I | A027750 |     |       |         |  1        |  1 2        |
%e A340035 | S | A027750 |     |       |         |  1        |  1 2        |
%e A340035 | O |---------|-----|-------|---------|-----------|-------------|
%e A340035 | R | A027750 |     |       |  1      |  1 2      |  1   3      |
%e A340035 | S | A027750 |     |       |  1      |  1 2      |  1   3      |
%e A340035 |   |---------|-----|-------|---------|-----------|-------------|
%e A340035 |   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |
%e A340035 |   |---------|-----|-------|---------|-----------|-------------|
%e A340035 |   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 .
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 |   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
%e A340035 |   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
%e A340035 | L | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
%e A340035 | I |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
%e A340035 | N | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
%e A340035 | K |         |  |  |  |\|  |  |\|\|  |  |\|\|\|  |  |\|\|\|\|  |
%e A340035 |   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 .
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 | P |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
%e A340035 | A |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
%e A340035 | R |         |     |       |  3      |  3 1      |  3 1 1      |
%e A340035 | T |         |     |       |         |  2 2      |  2 2 1      |
%e A340035 | I |         |     |       |         |  4        |  4 1        |
%e A340035 | T |         |     |       |         |           |  3 2        |
%e A340035 | I |         |     |       |         |           |  5          |
%e A340035 | O |         |     |       |         |           |             |
%e A340035 | N |         |     |       |         |           |             |
%e A340035 | S |         |     |       |         |           |             |
%e A340035 |---|---------|-----|-------|---------|-----------|-------------|
%e A340035 .
%e A340035 The table is essentially the same table of A340032 but here, in the upper zone, every row is A027750 instead of A127093.
%e A340035 Also the above table is the table of A338156 upside down.
%e A340035 The connection with the tower described in A221529 is as follows (n = 7):
%e A340035 |--------|------------------------|
%e A340035 | Level  |                        |
%e A340035 | in the | 7th slice of divisors  |
%e A340035 | tower  |                        |
%e A340035 |--------|------------------------|
%e A340035 |  11    |   1,                   |
%e A340035 |  10    |   1,                   |
%e A340035 |   9    |   1,                   |
%e A340035 |   8    |   1,                   |
%e A340035 |   7    |   1,                   |
%e A340035 |   6    |   1,                   |
%e A340035 |   5    |   1,                   |
%e A340035 |   4    |   1,                   |
%e A340035 |   3    |   1,                   |
%e A340035 |   2    |   1,                   |
%e A340035 |   1    |   1,                   |
%e A340035 |--------|------------------------|
%e A340035 |   7    |   1, 2,                |
%e A340035 |   6    |   1, 2,                |
%e A340035 |   5    |   1, 2,                |
%e A340035 |   4    |   1, 2,                |
%e A340035 |   3    |   1, 2,                |
%e A340035 |   2    |   1, 2,                |
%e A340035 |   1    |   1, 2,                |
%e A340035 |--------|------------------------|
%e A340035 |   5    |   1,    3,             |
%e A340035 |   4    |   1,    3,             |
%e A340035 |   3    |   1,    3,             |
%e A340035 |   2    |   1,    3,             |      Level
%e A340035 |   1    |   1,    3,             |             _
%e A340035 |--------|------------------------|       11   | |
%e A340035 |   3    |   1, 2,    4,          |       10   | |
%e A340035 |   2    |   1, 2,    4,          |        9   | |
%e A340035 |   1    |   1, 2,    4,          |        8   |_|_
%e A340035 |--------|------------------------|        7   |   |
%e A340035 |   2    |   1,          5,       |        6   |_ _|_
%e A340035 |   1    |   1,          5,       |        5   |   | |
%e A340035 |--------|------------------------|        4   |_ _|_|_
%e A340035 |   1    |   1, 2, 3,       6,    |        3   |_ _ _| |_
%e A340035 |--------|------------------------|        2   |_ _ _|_ _|_ _
%e A340035 |   1    |   1,                7; |        1   |_ _ _ _|_|_ _|
%e A340035 |--------|------------------------|
%e A340035              Figure 1.                            Figure 2.
%e A340035                                                 Lateral view
%e A340035                                                 of the tower.
%e A340035 .
%e A340035                                                 _ _ _ _ _ _ _
%e A340035                                                |_| | | | |   |
%e A340035                                                |_ _|_| | |   |
%e A340035                                                |_ _|  _|_|   |
%e A340035                                                |_ _ _|    _ _|
%e A340035                                                |_ _ _|  _|
%e A340035                                                |       |
%e A340035                                                |_ _ _ _|
%e A340035 .
%e A340035                                                   Figure 3.
%e A340035                                                   Top view
%e A340035                                                 of the tower.
%e A340035 .
%e A340035 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).
%t A340035 A340035row[n_]:=Flatten[Array[ConstantArray[Divisors[#],PartitionsP[n-#]]&,n]];
%t A340035 nrows=7;Array[A340035row,nrows] (* _Paolo Xausa_, Jun 20 2022 *)
%Y A340035 Nonzero terms of A340032.
%Y A340035 Row lengths give A006128, n >= 1.
%Y A340035 Row sums give A066186, n >= 1.
%Y A340035 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.
%K A340035 nonn,tabf
%O A340035 1,4
%A A340035 _Omar E. Pol_, Dec 26 2020

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