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A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.

%I A340531 #48 Jul 27 2021 01:38:36
%S A340531 1,4,1,8,4,1,1,15,8,4,4,1,1,1,21,15,8,8,4,4,4,1,1,1,1,1,33,21,15,15,8,
%T A340531 8,8,4,4,4,4,4,1,1,1,1,1,1,1,41,33,21,21,15,15,15,8,8,8,8,8,4,4,4,4,4,
%U A340531 4,4,1,1,1,1,1,1,1,1,1,1,1,56,41,33,33,21,21,21,15,15,15,15,15
%N A340531 Irregular triangle read by rows T(n,k), (n >= 1, k >= 1), in which row n has length is A000070(n-1) and every column k is A024916, the sum of all divisors of all numbers <= n.
%C A340531 Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593).
%C A340531 The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).
%C A340531 T(n,k) is the volume (the number of cells) in the k-th level starting from the base.
%C A340531 This polycube has the property that the volume (the total number of cells) equals A182738(n), the sum of all parts of all partitions of all positive integers <= n.
%C A340531 A dissection of the symmetric tower is a three-dimensional spiral whose top view is described in A239660.
%C A340531 Other triangles related to the volume of this polycube are A340527 and A340579.
%C A340531 The symmetric tower is a member of the family of the stepped pyramid described in A245092.
%C A340531 For another symmetric tower of the same family and whose volume equals A066186(n) see A340423.
%C A340531 The sum of row n of triangle equals A182738(n). That property is due to the correspondence between divisors and parts. For more information see A336811.
%F A340531 a(m) = A024916(A176206(m)), assuming A176206 has offset 1.
%F A340531 T(n,k) = A024916(A176206(n,k)), assuming A176206 has offset 1.
%e A340531 Triangle begins:
%e A340531    1;
%e A340531    4,  1;
%e A340531    8,  4,  1,  1;
%e A340531   15,  8,  4,  4, 1, 1, 1;
%e A340531   21, 15,  8,  8, 4, 4, 4, 1, 1, 1, 1, 1;
%e A340531   33, 21, 15, 15, 8, 8, 8, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1;
%e A340531 ...
%e A340531 For n = 5 the length of row 5 is A000070(4) = 12.
%e A340531 The sum of row 5 is 21 + 15 + 8 + 8 + 4 + 4 + 4 + 1 + 1 + 1 + 1 + 1 = 69, equaling A182738(5).
%Y A340531 Row sums give A182738.
%Y A340531 Cf. A340527 (a regular version).
%Y A340531 Members of the same family are: A176206, A337209, A339258, A340530.
%Y A340531 Cf. A221529, A239660, A339278, A339304, A340423, A340529.
%Y A340531 Cf. A000070, A024916, A237593, A336811, A338156.
%K A340531 nonn,tabf
%O A340531 1,2
%A A340531 _Omar E. Pol_, Jan 10 2021