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%I A338156 #143 May 29 2025 21:16:12
%S A338156 1,1,2,1,1,3,1,2,1,1,1,2,4,1,3,1,2,1,2,1,1,1,1,5,1,2,4,1,3,1,3,1,2,1,
%T A338156 2,1,2,1,1,1,1,1,1,2,3,6,1,5,1,2,4,1,2,4,1,3,1,3,1,3,1,2,1,2,1,2,1,2,
%U A338156 1,2,1,1,1,1,1,1,1,1,7,1,2,3,6,1,5,1,5,1,2,4,1,2,4,1,2,4
%N A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.
%C A338156 In other words: in row n replace every term of n-th row of A176206 with its divisors.
%C A338156 The terms in row n are also all parts of all partitions of n.
%C A338156 As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
%C A338156 For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
%C A338156 From _Omar E. Pol_, Aug 01 2021: (Start)
%C A338156 The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.
%C A338156 The terms of row n in nonincreasing order give the n-th row of A302246.
%C A338156 The terms of row n in nondecreasing order give the n-th row of A302247.
%C A338156 For the connection with the tower described in A221529 see also A340035. (End)
%H A338156 Paolo Xausa, <a href="/A338156/b338156.txt">Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)</a>
%e A338156 Triangle begins:
%e A338156 [1];
%e A338156 [1,2], [1];
%e A338156 [1,3], [1,2], [1], [1];
%e A338156 [1,2,4], [1,3], [1,2], [1,2], [1], [1], [1];
%e A338156 [1,5], [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
%e A338156 ...
%e A338156 For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
%e A338156 Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
%e A338156 [1],
%e A338156 -------
%e A338156 [1, 2],
%e A338156 [1],
%e A338156 -------
%e A338156 [1, 3],
%e A338156 [1, 2],
%e A338156 [1],
%e A338156 [1];
%e A338156 ----------
%e A338156 [1, 2, 4],
%e A338156 [1, 3],
%e A338156 [1, 2],
%e A338156 [1, 2],
%e A338156 [1],
%e A338156 [1],
%e A338156 [1];
%e A338156 ----------
%e A338156 [1, 5],
%e A338156 [1, 2, 4],
%e A338156 [1, 3],
%e A338156 [1, 3],
%e A338156 [1, 2],
%e A338156 [1, 2],
%e A338156 [1, 2],
%e A338156 [1],
%e A338156 [1],
%e A338156 [1],
%e A338156 [1],
%e A338156 [1];
%e A338156 .
%e A338156 The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
%e A338156 The table is infinite. It is formed by three zones as follows:
%e A338156 The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
%e A338156 The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
%e A338156 Finally the middle zone shows the connection between the upper zone and the lower zone.
%e A338156 For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
%e A338156 .
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 | n | | 1 | 2 | 3 | 4 | 5 |
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 | P | | | | | | |
%e A338156 | A | | | | | | |
%e A338156 | R | | | | | | |
%e A338156 | T | | | | | | 5 |
%e A338156 | I | | | | | | 3 2 |
%e A338156 | T | | | | | 4 | 4 1 |
%e A338156 | I | | | | | 2 2 | 2 2 1 |
%e A338156 | O | | | | 3 | 3 1 | 3 1 1 |
%e A338156 | N | | | 2 | 2 1 | 2 1 1 | 2 1 1 1 |
%e A338156 | S | | 1 | 1 1 | 1 1 1 | 1 1 1 1 | 1 1 1 1 1 |
%e A338156 ----|---------|-----|-------|---------|------------|---------------|
%e A338156 .
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 | | A181187 | 1 | 3 1 | 6 2 1 | 12 5 2 1 | 20 8 4 2 1 |
%e A338156 | | | | | |/| | |/|/| | |/ |/|/| | |/ | /|/|/| |
%e A338156 | L | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 |
%e A338156 | I | | * | * * | * * * | * * * * | * * * * * |
%e A338156 | N | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 |
%e A338156 | K | | = | = = | = = = | = = = = | = = = = = |
%e A338156 | | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 |
%e A338156 | | | | | |\| | |\|\| | |\ |\|\| | |\ |\ |\|\| |
%e A338156 | | A206561 | 1 | 4 2 | 9 5 3 | 20 13 7 4 | 35 23 15 9 5 |
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 .
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 | | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 |
%e A338156 | |---------|-----|-------|---------|------------|---------------|
%e A338156 | | A027750 | | 1 | 1 2 | 1 3 | 1 2 4 |
%e A338156 | |---------|-----|-------|---------|------------|---------------|
%e A338156 | D | A027750 | | | 1 | 1 2 | 1 3 |
%e A338156 | I | A027750 | | | 1 | 1 2 | 1 3 |
%e A338156 | V |---------|-----|-------|---------|------------|---------------|
%e A338156 | I | A027750 | | | | 1 | 1 2 |
%e A338156 | S | A027750 | | | | 1 | 1 2 |
%e A338156 | O | A027750 | | | | 1 | 1 2 |
%e A338156 | R |---------|-----|-------|---------|------------|---------------|
%e A338156 | S | A027750 | | | | | 1 |
%e A338156 | | A027750 | | | | | 1 |
%e A338156 | | A027750 | | | | | 1 |
%e A338156 | | A027750 | | | | | 1 |
%e A338156 | | A027750 | | | | | 1 |
%e A338156 |---|---------|-----|-------|---------|------------|---------------|
%e A338156 .
%e A338156 Note that every row in the lower zone lists A027750.
%e A338156 Also the lower zone for every positive integer can be constructed using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5]) then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
%e A338156 Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
%e A338156 The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
%e A338156 The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.
%t A338156 A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
%t A338156 A338156[10] (* Generates 10 rows *) (* _Paolo Xausa_, Jan 12 2023 *)
%o A338156 (PARI)
%o A338156 A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
%o A338156 A338156(10) \\ Generates 10 rows - _Paolo Xausa_, Feb 17 2023
%Y A338156 Nonzero terms of A340031.
%Y A338156 Row n has length A006128(n).
%Y A338156 The sum of row n is A066186(n).
%Y A338156 The product of row n is A007870(n).
%Y A338156 Row n lists the first n rows of A336812 (a subsequence).
%Y A338156 The number of parts k in row n is A066633(n,k).
%Y A338156 The sum of all parts k in row n is A138785(n,k).
%Y A338156 The number of parts >= k in row n is A181187(n,k).
%Y A338156 The sum of all parts >= k in row n is A206561(n,k).
%Y A338156 The number of parts <= k in row n is A210947(n,k).
%Y A338156 The sum of all parts <= k in row n is A210948(n,k).
%Y A338156 Cf. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.
%K A338156 nonn,tabf
%O A338156 1,3
%A A338156 _Omar E. Pol_, Oct 14 2020