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%I A339278 #98 Jan 19 2024 07:09:46
%S A339278 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,
%T A339278 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,
%U A339278 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
%N 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.
%C A339278 The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.
%C A339278 T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - _Omar E. Pol_, Jan 20 2022
%H A339278 Paolo Xausa, <a href="/A339278/b339278.txt">Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened)</a>
%F A339278 a(m) = A000203(A336811(m)).
%F A339278 T(n,k) = A000203(A336811(n,k)).
%e A339278 Triangle begins:
%e A339278 1;
%e A339278 3;
%e A339278 4, 1;
%e A339278 7, 3, 1;
%e A339278 6, 4, 3, 1, 1;
%e A339278 12, 7, 4, 3, 3, 1, 1;
%e A339278 8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1;
%e A339278 15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
%e A339278 13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
%e A339278 ...
%e A339278 From _Omar E. Pol_, Jan 13 2022: (Start)
%e A339278 Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
%e A339278 Level k: 1 2 3 4 5 6 7
%e A339278 Stage
%e A339278 n _ _ _ _ _ _ _ _
%e A339278 | _ |
%e A339278 1 | |_| |
%e A339278 |_ _ _ _ _ _ _ _|
%e A339278 | _ |
%e A339278 | | |_ |
%e A339278 2 | |_ _| |
%e A339278 |_ _ _ _ _ _ _ _|_ _ _ _ _ _
%e A339278 | _ | _ |
%e A339278 | | | | |_| |
%e A339278 3 | |_|_ _ | |
%e A339278 | |_ _| | |
%e A339278 |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _
%e A339278 | _ | _ | _ |
%e A339278 | | | | | |_ | |_| |
%e A339278 4 | | |_ | |_ _| | |
%e A339278 | |_ |_ _ | | |
%e A339278 | |_ _ _| | | |
%e A339278 |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
%e A339278 | _ | _ | _ | _ | _ |
%e A339278 | | | | | | | | |_ | |_| | |_| |
%e A339278 | | | | |_|_ _ | |_ _| | | |
%e A339278 5 | |_|_ | |_ _| | | | |
%e A339278 | |_ _ _ | | | | |
%e A339278 | |_ _ _| | | | | |
%e A339278 |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
%e A339278 | _ | _ | _ | _ | _ | _ | _ |
%e A339278 | | | | | | | | | | | |_ | | |_ | |_| | |_| |
%e A339278 | | | | | |_ | |_|_ _ | |_ _| | |_ _| | | |
%e A339278 | | |_ _ | |_ |_ _ | |_ _| | | | | |
%e A339278 6 | |_ | | |_ _ _| | | | | | |
%e A339278 | |_ |_ _ _ | | | | | | |
%e A339278 | |_ _ _ _| | | | | | | |
%e A339278 |_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
%e A339278 .
%e A339278 Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
%e A339278 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.
%e A339278 For n = 9 another connection with the tower is as follows:
%e A339278 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:
%e A339278 .
%e A339278 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e A339278 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
%e A339278 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;
%e A339278 7, 7, 7, 7, 7, 7, 7;
%e A339278 6, 6, 6, 6, 6;
%e A339278 12, 12, 12;
%e A339278 8, 8;
%e A339278 15;
%e A339278 13;
%e A339278 .
%e A339278 Then we rotate the triangle by 90 degrees as shown below:
%e A339278 _
%e A339278 1; | |
%e A339278 1; | |
%e A339278 1; | |
%e A339278 1; | |
%e A339278 1; | |
%e A339278 1; | |
%e A339278 1; |_|_
%e A339278 1, 3; | |
%e A339278 1, 3; | |
%e A339278 1, 3; | |
%e A339278 1, 3; |_ _|_
%e A339278 1, 3, 4; | | |
%e A339278 1, 3, 4; | | |
%e A339278 1, 3, 4; | | |
%e A339278 1, 3, 4; |_ _|_|_
%e A339278 1, 3, 4, 7; | | |
%e A339278 1, 3, 4, 7; |_ _ _| |_
%e A339278 1, 3, 4, 7, 6; | | |
%e A339278 1, 3, 4, 7, 6; |_ _ _|_ _|_
%e A339278 1, 3, 4, 7, 6, 12; |_ _ _ _| | |_
%e A339278 1, 3, 4, 7, 6, 12, 8; |_ _ _ _|_|_ _|_ _
%e A339278 1, 3, 4, 7, 6, 12, 8, 15; 13; |_ _ _ _ _|_ _|_ _|
%e A339278 .
%e A339278 Lateral view
%e A339278 of the tower
%e A339278 . _ _ _ _ _ _ _ _ _
%e A339278 |_| | | | | | | |
%e A339278 |_ _|_| | | | | |
%e A339278 |_ _| _|_| | | |
%e A339278 |_ _ _| _|_| |
%e A339278 |_ _ _| _| _ _|
%e A339278 |_ _ _ _| |
%e A339278 |_ _ _ _| _ _|
%e A339278 | |
%e A339278 |_ _ _ _ _|
%e A339278 .
%e A339278 Top view
%e A339278 of the tower
%e A339278 .
%e A339278 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)
%t A339278 A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1,n-m],PartitionsP[m]-PartitionsP[m-1]],{m,0,n-1}]],{n,rowmax}];
%t A339278 A339278[15] (* Generates 15 rows *) (* _Paolo Xausa_, Feb 17 2023 *)
%o A339278 (PARI) f(n) = numbpart(n-1);
%o A339278 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);}
%o A339278 tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ _Michel Marcus_, Jan 13 2021
%o A339278 (PARI) A339278(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(m-1)-numbpart(m-2),i,sigma(n-m+1)))));
%o A339278 A339278(15) \\ Generates 15 rows \\ _Paolo Xausa_, Feb 17 2023
%Y A339278 Sum of divisors of A336811.
%Y A339278 Row n has length A000041(n-1).
%Y A339278 Every column gives A000203.
%Y A339278 The length of the m-th block in row n is A187219(m), m >= 1.
%Y A339278 Row sums give A138879.
%Y A339278 Cf. A337209 (another version).
%Y A339278 Cf. A272172 (analog for the stepped pyramid described in A245092).
%Y A339278 Cf. A024916, A135010, A138121, A138137, A138785, A221529, A235791, A236104, A237270, A237591, A237593, A238442, A338156, A340423, A345023.
%K A339278 nonn,tabf
%O A339278 1,2
%A A339278 _Omar E. Pol_, Nov 29 2020