A339278 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which the partition number A000041(n-1) is the length of row n and every column k is A000203, the sum of divisors function.
1, 3, 4, 1, 7, 3, 1, 6, 4, 3, 1, 1, 12, 7, 4, 3, 3, 1, 1, 8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1, 15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 18, 15, 8, 12, 12, 6, 6, 7, 7, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Examples
Triangle begins:
1;
3;
4, 1;
7, 3, 1;
6, 4, 3, 1, 1;
12, 7, 4, 3, 3, 1, 1;
8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1;
15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From _Omar E. Pol_, Jan 13 2022: (Start)
Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
Level k: 1 2 3 4 5 6 7
Stage
n _ _ _ _ _ _ _ _
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1 | |_| |
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2 | |_ _| |
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3 | |_|_ _ | |
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4 | | |_ | |_ _| | |
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|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
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| | | | |_|_ _ | |_ _| | | |
5 | |_|_ | |_ _| | | | |
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|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
| _ | _ | _ | _ | _ | _ | _ |
| | | | | | | | | | | |_ | | |_ | |_| | |_| |
| | | | | |_ | |_|_ _ | |_ _| | |_ _| | | |
| | |_ _ | |_ |_ _ | |_ _| | | | | |
6 | |_ | | |_ _ _| | | | | | |
| |_ |_ _ _ | | | | | | |
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|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
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Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
For n = 6 and k = 3 we add four cubes at 6th stage in the third level of the structure of the tower starting from the base so T(6,3) = 4.
For n = 9 another connection with the tower is as follows:
First we take the columns from the above triangle and build a new triangle in which all columns start at row 1 as shown below:
.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;
7, 7, 7, 7, 7, 7, 7;
6, 6, 6, 6, 6;
12, 12, 12;
8, 8;
15;
13;
.
Then we rotate the triangle by 90 degrees as shown below:
_
1; | |
1; | |
1; | |
1; | |
1; | |
1; | |
1; |_|_
1, 3; | |
1, 3; | |
1, 3; | |
1, 3; |_ _|_
1, 3, 4; | | |
1, 3, 4; | | |
1, 3, 4; | | |
1, 3, 4; |_ _|_|_
1, 3, 4, 7; | | |
1, 3, 4, 7; |_ _ _| |_
1, 3, 4, 7, 6; | | |
1, 3, 4, 7, 6; |_ _ _|_ _|_
1, 3, 4, 7, 6, 12; |_ _ _ _| | |_
1, 3, 4, 7, 6, 12, 8; |_ _ _ _|_|_ _|_ _
1, 3, 4, 7, 6, 12, 8, 15; 13; |_ _ _ _ _|_ _|_ _|
.
Lateral view
of the tower
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|_ _| _|_| | | |
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Top view
of the tower
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The sum of the m-th row of the new triangle equals A024916(j) where j is the length of the m-th row, equaling the number of cubic cells in the m-th level of the tower. For example: the last row of triangle has 9 terms and the sum of the last row is 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 13 = A024916(9) = 69, equaling the number of cubes in the base of the tower. (End)
Links
Crossrefs
Programs
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Mathematica
A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}]; A339278[15] (* Generates 15 rows *) (* Paolo Xausa, Feb 17 2023 *)
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PARI
f(n) = numbpart(n-1); T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--;); sigma(1+s);} tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021 -
PARI
A339278(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,sigma(n-m+1))))); A339278(15) \\ Generates 15 rows \\ Paolo Xausa, Feb 17 2023
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