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A261350 Triangle read by rows T(n,k) which is the mirror of A237591.

%I A261350 #31 Dec 31 2020 11:11:15
%S A261350 1,2,1,2,1,3,2,3,1,1,4,1,2,4,1,2,5,2,2,5,1,1,2,6,1,1,3,6,1,2,2,7,1,2,
%T A261350 3,7,2,1,3,8,1,1,2,3,8,1,1,2,3,9,1,1,2,4,9,1,2,2,3,10,1,2,2,4,10,2,1,
%U A261350 2,4,11,1,1,1,3,4,11,1,1,2,2,4,12,1,1,2,2,5,12,1,1,2,3,4,13,1,2,1,3,5,13,1,2,2,2,5,14
%N A261350 Triangle read by rows T(n,k) which is the mirror of A237591.
%C A261350 Row n has length A003056(n) hence column k starts in row A000217(k).
%C A261350 Row sums give A000027.
%C A261350 Right border gives A008619, n >= 1.
%C A261350 n is an odd prime if and only if T(n,r-1) = 1 + T(n-1,r-1) and T(n,k) = T(n-1,k) for the rest of the values of k, where r = A003056(n) is the number of elements in row n.
%C A261350 T(n,k) is also the length of the k-th segment in a zig-zag path on the first quadrant of the square grid, connecting the point (m, m) with the point (0, n), ending with a segment in horizontal direction, where m = A240542(n). The area of the polygon defined by the y-axis, this zig-zag path and the diagonal [(0, 0), (m, m)], is equal to A024916(n)/2, one half of the sum of all divisors of all positive integers <= n. Therefore the reflected polygon, which is adjacent to the x-axis, with the zig-zag path connecting the point (n, 0) with the point (m, m), has the same property. And so on for each octant in the four quadrants.
%C A261350 For the representation of A024916 and A000203 we use two octants, for example: the first octant and the second octant, or the 6th octant and the 7th octant, etc., see A237593.
%C A261350 The elements of the n-th row of A237591 together with the elements of the n-th row of this sequence give the n-th row of A237593.
%C A261350 The connection between A196020 and A237271 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> this sequence --> A237593 --> A239660 --> A237270 --> A237271.
%C A261350 T(n,k) is also the area (or the number of cells) of the k-th vertical side at the n-th level (starting from the top) in the right part of the front view of the stepped pyramid described in A245092, see Example section.
%e A261350 Triangle begins:
%e A261350 Row
%e A261350 1                       1;
%e A261350 2                       2;
%e A261350 3                    1, 2;
%e A261350 4                    1, 3;
%e A261350 5                    2, 3;
%e A261350 6                 1, 1, 4;
%e A261350 7                 1, 2, 4;
%e A261350 8                 1, 2, 5;
%e A261350 9                 2, 2, 5;
%e A261350 10             1, 1, 2, 6;
%e A261350 11             1, 1, 3, 6;
%e A261350 12             1, 2, 2, 7;
%e A261350 13             1, 2, 3, 7;
%e A261350 14             2, 1, 3, 8;
%e A261350 15          1, 1, 2, 3, 8;
%e A261350 16          1, 1, 2, 3, 9;
%e A261350 17          1, 1, 2, 4, 9;
%e A261350 18          1, 2, 2, 3, 10;
%e A261350 19          1, 2, 2, 4, 10;
%e A261350 20          2, 1, 2, 4, 11;
%e A261350 21       1, 1, 1, 3, 4, 11;
%e A261350 22       1, 1, 2, 2, 4, 12;
%e A261350 23       1, 1, 2, 2, 5, 12;
%e A261350 24       1, 1, 2, 3, 4, 13;
%e A261350 25       1, 2, 1, 3, 5, 13;
%e A261350 26       1, 2, 2, 2, 5, 14;
%e A261350 ...
%e A261350 Illustration of initial terms:
%e A261350 Row      _
%e A261350 1       |1|_
%e A261350 2       |_ 2|_
%e A261350 3       |1|  2|_
%e A261350 4       |1|_   3|_
%e A261350 5       |_ 2|    3|_
%e A261350 6       |1|1|_     4|_
%e A261350 7       |1|  2|      4|_
%e A261350 8       |1|_ 2|_       5|_
%e A261350 9       |_ 2|  2|        5|_
%e A261350 10      |1|1|  2|_         6|_
%e A261350 11      |1|1|_   3|          6|_
%e A261350 12      |1|  2|  2|_           7|_
%e A261350 13      |1|_ 2|    3|            7|_
%e A261350 14      |_ 2|1|_   3|_             8|_
%e A261350 15      |1|1|  2|    3|              8|_
%e A261350 16      |1|1|  2|    3|_               9|_
%e A261350 17      |1|1|_ 2|_     4|                9|_
%e A261350 18      |1|  2|  2|    3|_                10|_
%e A261350 19      |1|_ 2|  2|      4|                 10|_
%e A261350 20      |_ 2|1|  2|_     4|_                  11|_
%e A261350 21      |1|1|1|_   3|      4|                   11|_
%e A261350 22      |1|1|  2|  2|      4|_                    12|_
%e A261350 23      |1|1|  2|  2|_       5|                     12|_
%e A261350 24      |1|1|_ 2|    3|      4|_                      13|_
%e A261350 25      |1|  2|1|_   3|        5|                       13|_
%e A261350 26      |1|  2|  2|  2|        5|                         14|
%e A261350 ...
%e A261350 Also the diagram represents the right part of the front view of the pyramid described in A245092. For the other half front view see A237591. For more information about the pyramid and the symmetric representation of sigma see A237593.
%Y A261350 Cf. A000203, A000217, A003056, A008016, A024916, A175254, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A240542, A244580, A245092, A249351, A259176, A259177, A259179.
%K A261350 nonn,tabf
%O A261350 1,2
%A A261350 _Omar E. Pol_, Aug 18 2015