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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.
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%I #111 Feb 04 2024 18:30:38

%S 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 30,36,16,19,51,52,77,42,60,24,30,23,57,68,91,66,84,40,45,26,29,69,76,

%U 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 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 From _Omar E. Pol_, Dec 21 2021: (Start)

%C Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.

%C 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 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 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 Ivan Neretin, <a href="/A272214/b272214.txt">Table of n, a(n) for n = 1..8128</a>

%F T(n,k) = prime(n)*sigma(k) = A000040(n)*A000203(k), n >= 1, k >= 1.

%F T(n,k) = A272400(n+1,k).

%e The corner of the square array begins:

%e 2, 6, 8, 14, 12, 24, 16, 30, 26, 36, ...

%e 3, 9, 12, 21, 18, 36, 24, 45, 39, 54, ...

%e 5, 15, 20, 35, 30, 60, 40, 75, 65, 90, ...

%e 7, 21, 28, 49, 42, 84, 56, 105, 91, 126, ...

%e 11, 33, 44, 77, 66, 132, 88, 165, 143, 198, ...

%e 13, 39, 52, 91, 78, 156, 104, 195, 169, 234, ...

%e 17, 51, 68, 119, 102, 204, 136, 255, 221, 306, ...

%e 19, 57, 76, 133, 114, 228, 152, 285, 247, 342, ...

%e 23, 69, 92, 161, 138, 276, 184, 345, 299, 414, ...

%e 29, 87, 116, 203, 174, 348, 232, 435, 377, 522, ...

%e ...

%e From _Omar E. Pol_, Dec 21 2021: (Start)

%e Written as a triangle the sequence begins:

%e 2;

%e 3, 6;

%e 5, 9, 8;

%e 7, 15, 12, 14;

%e 11, 21, 20, 21, 12;

%e 13, 33, 28, 35, 18, 24;

%e 17, 39, 44, 49, 30, 36, 16;

%e 19, 51, 52, 77, 42, 60, 24, 30;

%e 23, 57, 68, 91, 66, 84, 40, 45, 26;

%e 29, 69, 76, 119, 78, 132, 56, 75, 39, 36;

%e 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24;

%e ...

%e Row sums give A086718. (End)

%t Table[Prime[#] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* _Michael De Vlieger_, Apr 28 2016 *)

%Y Rows 1-4 of the square array: A074400, A272027, A274535, A319527.

%Y Columns 1-5 of the square array: A000040, A001748, A001749, A138636, A272470.

%Y Main diagonal of the square array gives A272211.

%Y Cf. A086718 (antidiagonal sums of the square array, row sums of the triangle).

%Y Cf. A000203, A221529, A237270, A237271, A237593, A245092, A272173, A272400, A274824.

%K nonn,tabl

%O 1,1

%A _Omar E. Pol_, Apr 28 2016