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A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.

%I A272214 #111 Feb 04 2024 18:30:38
%S A272214 2,3,6,5,9,8,7,15,12,14,11,21,20,21,12,13,33,28,35,18,24,17,39,44,49,
%T A272214 30,36,16,19,51,52,77,42,60,24,30,23,57,68,91,66,84,40,45,26,29,69,76,
%U A272214 119,78,132,56,75,39,36,31,87,92,133,102,156,88,105,65,54,24,37,93,116,161,114,204,104,165,91,90,36,56
%N A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.
%C A272214 From _Omar E. Pol_, Dec 21 2021: (Start)
%C A272214 Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.
%C A272214 For a visualization of T(n,j) first consider a tower (a polycube) in which the terraces are the symmetric representation of sigma(j), for j = 1 to n, starting from the top, and the heights of the terraces are the first n prime numbers respectively starting from the base. Then T(n,j) can be represented with a set of A237271(j) right prisms of height A000040(n-j+1) since T(n,j) is also the total number of cubes that are exactly below the parts of the symmetric representation of sigma(j) in the tower.
%C A272214 The sum of the n-th row of triangle is A086718(n) equaling the volume of the tower whose largest side of the base is n and its total height is the n-th prime.
%C A272214 The tower is an member of the family of the stepped pyramids described in A245092 and of the towers described in A221529. That is an infinite family of symmetric polycubes whose volumes represent the convolution of A000203 with any other integer sequence. (End)
%H A272214 Ivan Neretin, <a href="/A272214/b272214.txt">Table of n, a(n) for n = 1..8128</a>
%F A272214 T(n,k) = prime(n)*sigma(k) = A000040(n)*A000203(k), n >= 1, k >= 1.
%F A272214 T(n,k) = A272400(n+1,k).
%e A272214 The corner of the square array begins:
%e A272214    2,  6,   8,  14,  12,  24,  16,  30,  26,  36, ...
%e A272214    3,  9,  12,  21,  18,  36,  24,  45,  39,  54, ...
%e A272214    5, 15,  20,  35,  30,  60,  40,  75,  65,  90, ...
%e A272214    7, 21,  28,  49,  42,  84,  56, 105,  91, 126, ...
%e A272214   11, 33,  44,  77,  66, 132,  88, 165, 143, 198, ...
%e A272214   13, 39,  52,  91,  78, 156, 104, 195, 169, 234, ...
%e A272214   17, 51,  68, 119, 102, 204, 136, 255, 221, 306, ...
%e A272214   19, 57,  76, 133, 114, 228, 152, 285, 247, 342, ...
%e A272214   23, 69,  92, 161, 138, 276, 184, 345, 299, 414, ...
%e A272214   29, 87, 116, 203, 174, 348, 232, 435, 377, 522, ...
%e A272214   ...
%e A272214 From _Omar E. Pol_, Dec 21 2021: (Start)
%e A272214 Written as a triangle the sequence begins:
%e A272214    2;
%e A272214    3,  6;
%e A272214    5,  9,  8;
%e A272214    7, 15, 12,  14;
%e A272214   11, 21, 20,  21,  12;
%e A272214   13, 33, 28,  35,  18,  24;
%e A272214   17, 39, 44,  49,  30,  36, 16;
%e A272214   19, 51, 52,  77,  42,  60, 24,  30;
%e A272214   23, 57, 68,  91,  66,  84, 40,  45, 26;
%e A272214   29, 69, 76, 119,  78, 132, 56,  75, 39, 36;
%e A272214   31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24;
%e A272214 ...
%e A272214 Row sums give A086718. (End)
%t A272214 Table[Prime[#] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Apr 28 2016 *)
%Y A272214 Rows 1-4 of the square array: A074400, A272027, A274535, A319527.
%Y A272214 Columns 1-5 of the square array: A000040, A001748, A001749, A138636, A272470.
%Y A272214 Main diagonal of the square array gives A272211.
%Y A272214 Cf. A086718 (antidiagonal sums of the square array, row sums of the triangle).
%Y A272214 Cf. A000203, A221529, A237270, A237271, A237593, A245092, A272173, A272400, A274824.
%K A272214 nonn,tabl
%O A272214 1,1
%A A272214 _Omar E. Pol_, Apr 28 2016