Displaying 1-10 of 49 results found.
Irregular triangle read by rows T(n,k), n >= 1, k >= 1, in which row n is constructed replacing every term of row n of A336811 with its divisors.
+20
27
1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 4, 8, 1, 2, 3, 6, 1, 5, 1, 2, 4, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 9, 1, 7, 1, 2, 3, 6
COMMENTS
Here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the corresponce between all parts of the last section of the set of partitions of n and all divisors of all terms of the n-th row of A336811, with n >= 1. The mentionded parts and the mentioned divisors are the same numbers (see Example section). For an equivalent table showing the same kind of correspondence for all partitions of all positive integers see the supersequence A338156.
EXAMPLE
Triangle begins:
[1];
[1, 2];
[1, 3], [1];
[1, 2, 4], [1, 2], [1];
[1, 5], [1, 3], [1, 2], [1], [1];
[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1];
...
For n = 6 the 6th row of A336811 is [6, 4, 3, 2, 2, 1, 1] so replacing every term with its divisors we have {[1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]} the same as the 6th row of this triangle. Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
-------------
[1],
-------------
[1, 2];
-------------
[1, 3],
[1];
-------------
[1, 2, 4],
[1, 2],
[1];
-------------
[1, 5],
[1, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 3, 6],
[1, 2, 4],
[1, 3],
[1, 2],
[1, 2],
[1],
[1];
-------------
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and the parts of the last section of the set of partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the last section of the set of partitions of every positive integer.
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n | | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| | | | | | | | 6 |
| P | | | | | | | 3 3 |
| A | | | | | | | 4 2 |
| R | | | | | | | 2 2 2 |
| T | | | | | | 5 | 1 |
| I | | | | | | 3 2 | 1 |
| T | | | | | 4 | 1 | 1 |
| I | | | | | 2 2 | 1 | 1 |
| O | | | | 3 | 1 | 1 | 1 |
| N | | | 2 | 1 | 1 | 1 | 1 |
| S | | 1 | 1 | 1 | 1 | 1 | 1 |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| | A207031 | 1 | 2 1 | 3 1 1 | 6 3 1 1 | 8 3 2 1 1 | 15 8 4 2 1 1 | | L | | | | |/| | |/|/| | |/|/|/| | |/|/|/|/| | |/|/|/|/|/| |
| I | A182703 | 1 | 1 1 | 2 0 1 | 3 2 0 1 | 5 1 1 0 1 | 7 4 2 1 0 1 | | N | | * | * * | * * * | * * * * | * * * * * | * * * * * * |
| K | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 | 1 2 3 4 5 6 | | | | = | = = | = = = | = = = = | = = = = = | = = = = = = |
| | A207383 | 1 | 1 2 | 2 0 3 | 3 4 0 4 | 5 2 3 0 5 | 7 8 6 4 0 6 | |---|---------|-----|-------|---------|-----------|-------------|---------------|
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| | A027750 | 1 | 1 2 | 1 3 | 1 2 4 | 1 5 | 1 2 3 6 | | D |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 | | | 1 | 1 2 | 1 3 | 1 2 4 | | V |---------|-----|-------|---------|-----------|-------------|---------------|
| I | A027750 | | | | 1 | 1 2 | 1 3 | | S |---------|-----|-------|---------|-----------|-------------|---------------|
| S |---------|-----|-------|---------|-----------|-------------|---------------|
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
Note that every row in the lower zone lists A027750. The "section" is the simpler substructure of the set of partitions of n that has this property in the three zones.
Also the lower zone for every positive integer can be constructued using the first n terms of A002865. For example: for n = 6 we consider the first 6 terms of A002865 (that is [1, 0, 1, 1, 2, 2] and then the 6th slice is formed by a block with the divisors of 6, no block with the divisors of 5, one block with the divisors of 4, one block with the divisors of 3, two block with the divisors of 2 and two blocks with the divisors of 1. 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. 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). The above table shows the growth step by step of both the prism of partitions and its associated tower since the number of parts in the last section of the set of partitions of n is equal to A138137(n) equaling the number of divisors in the n-th slice of the lower table and equalimg the same the number of terms in the n-th row of triangle. Also the sum of all parts in the last section of the set of partitions of n is equal to A138879(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.
MATHEMATICA
A336812[row_]:=Flatten[Table[ConstantArray[Divisors[row-m], PartitionsP[m]-PartitionsP[m-1]], {m, 0, row-1}]];
CROSSREFS
Companion and subsequence of A338156. Cf. A000041, A000070, A002260, A002865, A006128, A024916, A027750, A066186, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A187219, A207031, A207038, A207383, A221529, A221530, A221531, A237593, A245095, A221649, A221650, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A350357.
Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.
+20
4
1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1
COMMENTS
The terms in row n are also all parts of all partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246. The terms of row n in nondecreasing order give the n-th row of A302247. For further information about the correspondence divisor/part see A336811 and A338156.
EXAMPLE
Triangle begins:
[1];
[1],[1, 2];
[1],[1, 2],[1, 3],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
...
Below the table shows the correspondence divisor/part.
|---|-----------------|-----|-------|---------|-----------|-------------|
| 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 | |---|-----------------|-----|-------|---------|-----------|-------------|
.
. |-------|
. |Section|
|---|-------|---------|-----|-------|---------|-----------|-------------|
| | 1 | A000012 | 1 | 1 | 1 | 1 | 1 | | |-------|---------|-----|-------|---------|-----------|-------------|
| | 2 | A000034 | | 1 2 | 1 2 | 1 2 | 1 2 | | |-------|---------|-----|-------|---------|-----------|-------------|
| D | 3 | A010684 | | | 1 3 | 1 3 | 1 3 | | V |-------|---------|-----|-------|---------|-----------|-------------|
| I | 4 | A069705 | | | | 1 2 4 | 1 2 4 | | S | | A000034 | | | | 1 2 | 1 2 | | R |-------|---------|-----|-------|---------|-----------|-------------|
|---|-------|---------|-----|-------|---------|-----------|-------------|
.
In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n. The number of rows in the j-th section of the lower zone is equal to A000041(j-1). The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.
CROSSREFS
The product of row n is A007870(n). Row n lists the first n rows of A336812. The number of parts k in row n is A066633(n,k). The sum of all parts k in row n is A138785(n,k). The number of parts >= k in row n is A181187(n,k). The sum of all parts >= k in row n is A206561(n,k). The number of parts <= k in row n is A210947(n,k). The sum of all parts <= k in row n is A210948(n,k). Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.
Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: row n is constructed by replacing A336811(n,k) with the largest partition into consecutive parts of A000217( A336811(n,k)).
+20
2
1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 1, 2, 1, 1, 5, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1, 6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1, 7, 6, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 8, 7, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1
COMMENTS
All divisors of all terms in row n are also all parts of all partitions of n.
The terms of row n listed in nonincreasing order give the n-th row of A176206. The number of k's in row n is equal to A000041(n-k), 1 <= k <= n. The number of terms >= k in row n is equal to A000070(n-k), 1 <= k <= n. The number of k's in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n. The number of terms >= k in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A014153(n-k), 1 <= k <= n. Row n is constructed replacing A336811(n,k) with the largest partition into consecutive parts of A359279(n,k).
EXAMPLE
Triangle begins:
1;
2, 1;
3, 2, 1, 1;
4, 3, 2, 1, 2, 1, 1;
5, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1;
6, 5, 4, 3, 2, 1, 4, 3, 2, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1;
...
Or also the triangle begins:
[1];
[2, 1];
[3, 2, 1], [1];
[4, 3, 2, 1], [2, 1], [1];
[5, 4, 3, 2, 1], [3, 2, 1], [2, 1], [1], [1];
[6, 5, 4, 3, 2, 1], [4, 3, 2, 1], [3, 2, 1], [2, 1], [2, 1], [1], [1];
...
For n = 3 the third row is [3, 2, 1, 1]. The divisors of these terms are [1, 3], [1, 2], [1], [1]. These six divisors are also all parts of all partitions of 3. They are [3], [2, 1], [1, 1, 1].
MATHEMATICA
A359350row[n_]:=Flatten[Table[ConstantArray[Range[n-m, 1, -1], PartitionsP[m]-PartitionsP[m-1]], {m, 0, n-1}]]; Array[A359350row, 10] (* Paolo Xausa, Sep 01 2023 *)
Irregular triangle read by rows T(n,k) in which row n lists the terms of n-th row of A336811 in nondecreasing order.
+20
1
1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 3, 4, 6, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 8, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 7, 9, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10
COMMENTS
All divisors of all terms of n-th row are also all parts of the last section of the set of partitions of n.
All divisors of all terms of the first n rows are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n. For further information about the correspondence divisor/part see A338156 and A336812.
EXAMPLE
Triangle begins:
1;
2;
1, 3;
1, 2, 4;
1, 1, 2, 3, 5;
1, 1, 2, 2, 3, 4, 6;
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7;
1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 8;
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 7, 9;
...
MATHEMATICA
A341049[rowmax_]:=Table[Flatten[Table[ConstantArray[n-m, PartitionsP[m]-PartitionsP[m-1]], {m, n-1, 0, -1}]], {n, rowmax}];
PROG
(PARI)
A341049(rowmax)=vector(rowmax, n, concat(vector(n, m, vector(numbpart(n-m)-numbpart(n-m-1), i, m))));
CROSSREFS
Cf. A000070, A000041, A002865, A027750, A028310, A133735, A135010, A138121, A138137, A176206, A182703, A187219, A207378, A237593, A336812, A338156, A339278, A340061.
1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133
COMMENTS
Number of parts in the last section of the set of partitions of n (see A135010, A138121). Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011 a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016 G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017
EXAMPLE
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
. 6
. 3+3
. 4+2
. 2+2+2
. 5 1
. 3+2 1
. 4 1 1
. 2+2 1 1
. 3 1 1 1
. 2 1 1 1 1
. 1 1 1 1 1 1
. --- ----- ------- --------- ----------- --------------
. 1, 2, 3, 6, 8, 15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End) The geometric model looks like this:
. _ _ _ _ _ _
. |_ _ _ _ _ _|
. |_ _ _|_ _ _|
. |_ _ _ _|_ _|
. _ _ _ _ _ |_ _|_ _|_ _|
. |_ _ _ _ _| |_|
. _ _ _ _ |_ _ _|_ _| |_|
. |_ _ _ _| |_| |_|
. _ _ _ |_ _|_ _| |_| |_|
. _ _ |_ _ _| |_| |_| |_|
. _ |_ _| |_| |_| |_| |_|
. |_| |_| |_| |_| |_| |_|
.
. 1 2 3 6 8 15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - Omar E. Pol, Jul 27 2021
MAPLE
b:= proc(n, i) option remember; local f, g;
if n=0 then [1, 0]
elif i<1 then [0, 0]
elif i>n then b(n, i-1)
else f:= b(n, i-1); g:= b(n-i, i);
[f[1]+g[1], f[2]+g[2] +g[1]]
fi
end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
MATHEMATICA
b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *) Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)
CROSSREFS
Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.
Sum of all parts of the last section of the set of partitions of n.
+10
50
1, 3, 5, 11, 15, 31, 39, 71, 94, 150, 196, 308, 389, 577, 750, 1056, 1353, 1881, 2380, 3230, 4092, 5412, 6821, 8935, 11150, 14386, 17934, 22834, 28281, 35735, 43982, 55066, 67551, 83821, 102365, 126267, 153397, 188001, 227645, 277305, 334383
COMMENTS
a(n) is also the sum of all divisors of all terms of n-th row of A336811. These divisors are also all parts in the last section of the set of partitions of n. - Omar E. Pol, Jul 27 2021
FORMULA
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi/(12*sqrt(2*n)) * (1 - (72 + 13*Pi^2) / (24*Pi*sqrt(6*n)) + (7/12 + 3/(2*Pi^2) + 217*Pi^2/6912)/n - (15*sqrt(3/2)/(16*Pi) + 115*Pi/(288*sqrt(6)) + 4069*Pi^3/(497664*sqrt(6)))/n^(3/2)). - Vaclav Kotesovec, Oct 21 2016, extended Jul 06 2019 G.f.: x*(1 - x)*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017
EXAMPLE
a(6)=31 because the parts of the last section of the set of partitions of 6 are (6), (3,3), (4,2), (2,2,2), (1), (1), (1), (1), (1), (1), (1), so the sum is a(6) = 6 + 3 + 3 + 4 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 31.
Illustration of initial terms:
. _ _ _ _ _ _
. |_ _ _ _ _ _|
. |_ _ _|_ _ _|
. |_ _ _ _|_ _|
. _ _ _ _ _ |_ _|_ _|_ _|
. |_ _ _ _ _| |_|
. _ _ _ _ |_ _ _|_ _| |_|
. |_ _ _ _| |_| |_|
. _ _ _ |_ _|_ _| |_| |_|
. _ _ |_ _ _| |_| |_| |_|
. _ |_ _| |_| |_| |_| |_|
. |_| |_| |_| |_| |_| |_|
.
. 1 3 5 11 15 31
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the sum of all divisors of these terms is [1 + 2 + 3 + 6] + [1 + 2 + 4] + [1 + 3] + [1 + 2] + [1 + 2] + [1] + [1] = 31, so a(6) = 31. - Omar E. Pol, Jul 27 2021
MAPLE
A066186 := proc(n) n*combinat[numbpart](n) ; end proc:
MATHEMATICA
Table[PartitionsP[n]*n - PartitionsP[n-1]*(n-1), {n, 1, 50}] (* Vaclav Kotesovec, Oct 21 2016 *)
PROG
(PARI) for(n=1, 50, print1(numbpart(n)*n - numbpart(n - 1)*(n - 1), ", ")) \\ Indranil Ghosh, Mar 19 2017 (Python)
from sympy.ntheory import npartitions
print([npartitions(n)*n - npartitions(n - 1)*(n - 1) for n in range(1, 51)]) # Indranil Ghosh, Mar 19 2017
CROSSREFS
Cf. A000041, A000203, A002865, A066186, A133041, A135010, A138121, A138135 - A138138, A138151, A138880, A139100, A237593, A336811, A336812, A338156, A339278, A340035, A340426, A340583, A340793.
1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28
COMMENTS
Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014 The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base. The base of the tower is the symmetric representation of A024916(n). The height of the tower is equal to A000041(n-1). The surface area of the tower is equal to A345023(n). The volume (or the number of cubes) of the tower equals A066186(n). Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1]. The tower is an object of the family of the stepped pyramid described in A245092. T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower. T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206. The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n. Therefore the set of all partitions of n >= 1 has an associated tower.
The partial column sums of A340583 give this triangle showing the growth of the structure of the tower. Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)
EXAMPLE
Triangle begins:
------------------------------------------------------
n| k 1 2 3 4 5 6 7 8 9 10
------------------------------------------------------
1| 1;
2| 1, 3;
3| 2, 3, 4;
4| 3, 6, 4, 7;
5| 5, 9, 8, 7, 6;
6| 7, 15, 12, 14, 6, 12;
7| 11, 21, 20, 21, 12, 12, 8;
8| 15, 33, 28, 35, 18, 24, 8, 15;
9| 22, 45, 44, 49, 30, 36, 16, 15, 13;
10| 30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
…
The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420. .
For n = 10 the calculation of the row 10 is as follows:
1 1 * 30 = 30
2 3 * 22 = 66
3 4 * 15 = 60
4 7 * 11 = 77
5 6 * 7 = 42
6 12 * 5 = 60
7 8 * 3 = 24
8 15 * 2 = 30
9 13 * 1 = 13
10 18 * 1 = 18
.
For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower". Both objects contain the same number of cubes (that property is valid for n >= 1).
_ _ _ _ _ _ _ _ _ _
42 |_ _ _ _ _ |
|_ _ _ _ _|_ |
|_ _ _ _ _ _|_ |
|_ _ _ _ | |
|_ _ _ _|_ _ _|_ |
|_ _ _ _ | |
|_ _ _ _|_ | |
|_ _ _ _ _|_ | |
|_ _ _ | | |
|_ _ _|_ | | |
|_ _ | | | |
|_ _|_ _|_ _|_ _|_ | _
30 |_ _ _ _ _ | | | | 30
|_ _ _ _ _|_ | | | |
|_ _ _ | | | | |
|_ _ _|_ _ _|_ | | | |
|_ _ _ _ | | | | |
|_ _ _ _|_ | | | | |
|_ _ _ | | | | | |
|_ _ _|_ _|_ _|_ | | _|_|
22 |_ _ _ _ | | | | | 22
|_ _ _ _|_ | | | | |
|_ _ _ _ _|_ | | | | |
|_ _ _ | | | | | |
|_ _ _|_ | | | | | |
|_ _ | | | | | | |
|_ _|_ _|_ _|_ | | | _|_ _|
15 |_ _ _ _ | | | | | | | 15
|_ _ _ _|_ | | | | | | |
|_ _ _ | | | | | | | |
|_ _ _|_ _|_ | | | | _|_|_ _|
11 |_ _ _ | | | | | | | | 11
|_ _ _|_ | | | | | | | |
|_ _ | | | | | | | | |
|_ _|_ _|_ | | | | | _| |_ _ _|
7 |_ _ _ | | | | | | | | | 7
|_ _ _|_ | | | | | | _|_ _|_ _ _|
5 |_ _ | | | | | | | | | | | 5
|_ _|_ | | | | | | | _| | |_ _ _ _|
3 |_ _ | | | | | | | | _|_ _|_|_ _ _ _| 3
2 |_ | | | | | | | | | _ _|_ _|_|_ _ _ _ _| 2
1 |_|_|_|_|_|_|_|_|_|_| |_ _|_|_|_ _ _ _ _ _| 1
.
Figure 1. Figure 2.
Front view of the Lateral view
prism of partitions. of the tower.
.
. _ _ _ _ _ _ _ _ _ _
| | | | | | | | |_| 1
| | | | | | |_|_ _| 2
| | | | |_|_ |_ _| 3
| | |_|_ |_ _ _| 4
| |_ _ |_ |_ _ _| 5
|_ _ |_ |_ _ _ _| 6
|_ | |_ _ _ _| 7
|_ |_ _ _ _ _| 8
| | 9
|_ _ _ _ _ _| 10
.
Figure 3.
Top view
of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 10 in colexicographic order (cf. A026792, A211992). The area of the diagram is 10*42 = A066186(10) = 420. Note that the diagram can be interpreted also as the front view of a right prism whose volume is 1*10*42 = 420 equaling the volume and the number of cubes of the tower that appears in the figures 2 and 3. Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions. In this case the mentioned area equals A000070(10-1) = 97. The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812. The sum of the volumes of both objects equals A220909.
MATHEMATICA
nrows=15; Table[Table[DivisorSigma[1, k]PartitionsP[n-k], {k, n}], {n, nrows}] (* Paolo Xausa *), Jun 17 2022
CROSSREFS
Cf. A000070, A000203, A026792, A027293, A135010, A138137, A176206, A182703, A220909, A211992, A221649, A236104, A237270, A237271, A237593, A245092, A245093, A245095, A245099, A262626, A336811, A336812, A338156, A339278, A340035, A340583, A340584, A345023, A346741.
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.
+10
36
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, 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, 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
COMMENTS
In other words: in row n replace every term of n-th row of A176206 with its divisors. The terms in row n are also all parts of all partitions of n.
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. 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. The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n. The terms of row n in nonincreasing order give the n-th row of A302246. The terms of row n in nondecreasing order give the n-th row of A302247. For the connection with the tower described in A221529 see also A340035. (End)
EXAMPLE
Triangle begins:
[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], [1,2], [1,2], [1], [1], [1], [1], [1];
...
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. Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
[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, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
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.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992). The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
.
|---|---------|-----|-------|---------|------------|---------------|
| 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 | A066633 | 1 | 2 1 | 4 1 1 | 7 3 1 1 | 12 4 2 1 1 | | I | | * | * * | * * * | * * * * | * * * * * |
| N | A002260 | 1 | 1 2 | 1 2 3 | 1 2 3 4 | 1 2 3 4 5 | | K | | = | = = | = = = | = = = = | = = = = = |
| | A138785 | 1 | 2 2 | 4 2 3 | 7 6 3 4 | 12 8 6 4 5 | | | | | | |\| | |\|\| | |\ |\|\| | |\ |\ |\|\| |
| | A206561 | 1 | 4 2 | 9 5 3 | 20 13 7 4 | 35 23 15 9 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 |---------|-----|-------|---------|------------|---------------|
|---|---------|-----|-------|---------|------------|---------------|
.
Note that every row in the lower zone lists A027750. Also the lower zone for every positive integer can be constructued 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. 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. 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). 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.
MATHEMATICA
A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m], PartitionsP[m]], {m, 0, n-1}]], {n, rowmax}];
PROG
(PARI)
A338156(rowmax)=vector(rowmax, n, concat(vector(n, m, concat(vector(numbpart(m-1), i, divisors(n-m+1))))));
CROSSREFS
The product of row n is A007870(n). Row n lists the first n rows of A336812 (a subsequence). The number of parts k in row n is A066633(n,k). The sum of all parts k in row n is A138785(n,k). The number of parts >= k in row n is A181187(n,k). The sum of all parts >= k in row n is A206561(n,k). The number of parts <= k in row n is A210947(n,k). The sum of all parts <= k in row n is A210948(n,k). 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.
Irregular triangle T(n,k) (n >= 1, k >= 1) read by rows: row n has length A000070(n-1) and every column k gives the positive integers.
+10
31
1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3
COMMENTS
The original definition was: An irregular table: Row n begins with n, counts down to 1 and repeats the intermediate numbers as often as given by the partition numbers.
Row n contains a decreasing sequence where n-k is repeated A000041(k) times, k = 0..n-1. Row n lists in nonincreasing order the first A000070(n-1) terms of A336811. In other words: row n lists in nonincreasing order the terms from the first n rows of triangle A336811. Conjecture: all divisors of all terms in row n are also all parts of all partitions of n.
For more information see the example and A336811 which contains the most elementary conjecture about the correspondence divisors/partitions. A338156 lists the divisors of every term of this sequence.
The n-th row of A340581 lists in nonincreasing order the terms of the first n rows of this triangle. For a regular triangle with the same row sums see A141157. (End) The number of k's in row n is equal to A000041(n-k), 1 <= k <= n. The number of terms >= k in row n is equal to A000070(n-k), 1 <= k <= n. The number of k's in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n. The number of terms >= k in the first n rows (or in the first A014153(n-1) terms of the sequence) is equal to A014153(n-k), 1 <= k <= n. (End)
EXAMPLE
Triangle begins:
1;
2, 1;
3, 2, 1, 1;
4, 3, 2, 2, 1, 1, 1;
5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1;
6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
7, 6, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, ...
For n = 5, by definition the length of row 5 is A000070(5-1) = A000070(4) = 12, so the row 5 of triangle has 12 terms. Since every column lists the positive integers A000027 so the row 5 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1]. Then we have that the divisors of the numbers of the 5th row are:
.
5th row of triangle -----> 5 4 3 3 2 2 2 1 1 1 1 1
1 2 1 1 1 1 1
1
.
There are twelve 1's, four 2's, two 3's, one 4 and one 5.
In total there are 12 + 4 + 2 + 1 + 1 = 20 divisors.
On the other hand the partitions of 5 are as shown below:
.
. 5
. 3 2
. 4 1
. 2 2 1
. 3 1 1
. 2 1 1 1
. 1 1 1 1 1
.
There are twelve 1's, four 2's, two 3's, one 4 and one 5, as shown also in the 5th row of triangle A066633. In total there are 12 + 4 + 2 + 1 + 1 = A006128(5) = 20 parts. Finally in accordance with the conjecture we can see that all divisors of all numbers in the 5th row of the triangle are the same positive integers as all parts of all partitions of 5. (End)
MATHEMATICA
Table[Flatten[Table[ConstantArray[n-k, PartitionsP[k]], {k, 0, n-1}]], {n, 10}] (* Paolo Xausa, May 30 2022 *)
EXTENSIONS
New name, changed offset, edited and more terms from Omar E. Pol, Nov 22 2020
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.
+10
26
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, 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, 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
COMMENTS
For further information about the correspondence divisor/part see A338156.
EXAMPLE
Triangle begins:
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, 3, 1, 3, 1, 2, 4, 1, 5;
...
Written as an irregular tetrahedron the first five slices are:
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, 3,
1, 3,
1, 2, 4,
1, 5;
--------
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):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n | | 1 | 2 | 3 | 4 | 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 | |---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
| | 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 | | | | | | |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A340032 but here, in the upper zone, every row is A027750 instead of A127093. Also the above table is the table of A338156 upside down. The connection with the tower described in A221529 is as follows (n = 7): |--------|------------------------|
| Level | |
| in the | 7th slice of divisors |
| tower | |
|--------|------------------------|
| 11 | 1, |
| 10 | 1, |
| 9 | 1, |
| 8 | 1, |
| 7 | 1, |
| 6 | 1, |
| 5 | 1, |
| 4 | 1, |
| 3 | 1, |
| 2 | 1, |
| 1 | 1, |
|--------|------------------------|
| 7 | 1, 2, |
| 6 | 1, 2, |
| 5 | 1, 2, |
| 4 | 1, 2, |
| 3 | 1, 2, |
| 2 | 1, 2, |
| 1 | 1, 2, |
|--------|------------------------|
| 5 | 1, 3, |
| 4 | 1, 3, |
| 3 | 1, 3, |
| 2 | 1, 3, | Level
| 1 | 1, 3, | _
|--------|------------------------| 11 | |
| 3 | 1, 2, 4, | 10 | |
| 2 | 1, 2, 4, | 9 | |
| 1 | 1, 2, 4, | 8 |_|_
|--------|------------------------| 7 | |
| 2 | 1, 5, | 6 |_ _|_
| 1 | 1, 5, | 5 | | |
|--------|------------------------| 4 |_ _|_|_
| 1 | 1, 2, 3, 6, | 3 |_ _ _| |_
|--------|------------------------| 2 |_ _ _|_ _|_ _
| 1 | 1, 7; | 1 |_ _ _ _|_|_ _|
|--------|------------------------|
Figure 1. Figure 2.
Lateral view
of the tower.
.
_ _ _ _ _ _ _
|_| | | | | |
|_ _|_| | | |
|_ _| _|_| |
|_ _ _| _ _|
|_ _ _| _|
| |
|_ _ _ _|
.
Figure 3.
Top view
of the tower.
.
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).
MATHEMATICA
A340035row[n_]:=Flatten[Array[ConstantArray[Divisors[#], PartitionsP[n-#]]&, n]];
nrows=7; Array[A340035row, nrows] (* Paolo Xausa, Jun 20 2022 *)
CROSSREFS
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.
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