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A340526 Triangle read by rows: T(n,k) = A006218(n-k+1)*A000041(k-1), 1 <= k <= n.

%I A340526 #37 Jul 25 2021 13:47:29
%S A340526 1,3,1,5,3,2,8,5,6,3,10,8,10,9,5,14,10,16,15,15,7,16,14,20,24,25,21,
%T A340526 11,20,16,28,30,40,35,33,15,23,20,32,42,50,56,55,45,22,27,23,40,48,70,
%U A340526 70,88,75,66,30,29,27,46,60,80,98,110,120,110,90,42,35,29,54,69,100,112,154,150,176,150,126,56
%N A340526 Triangle read by rows: T(n,k) = A006218(n-k+1)*A000041(k-1), 1 <= k <= n.
%C A340526 Conjecture 1: T(n,k) is the total number of divisors of the terms that are in the k-th blocks of the first n rows of triangle A176206.
%C A340526 Conjecture 2: the sum of row n equals A284870, the total number of parts in all partitions of all positive integers <= n.
%C A340526 The above conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).
%e A340526 Triangle begins:
%e A340526    1;
%e A340526    3,  1;
%e A340526    5,  3,  2;
%e A340526    8,  5,  6,  3;
%e A340526   10,  8, 10,  9,   5;
%e A340526   14, 10, 16, 15,  15,   7;
%e A340526   16, 14, 20, 24,  25,  21,  11;
%e A340526   20, 16, 28, 30,  40,  35,  33,  15;
%e A340526   23, 20, 32, 42,  50,  56,  55,  45,  22;
%e A340526   27, 23, 40, 48,  70,  70,  88,  75,  66,  30;
%e A340526   29, 27, 46, 60,  80,  98, 110, 120, 110,  90,  42;
%e A340526   35, 29, 54, 69, 100, 112, 154, 150, 176, 150, 126,  56;
%e A340526 ...
%e A340526 For n = 6 the calculation of every term of row 6 is as follows:
%e A340526 --------------------------
%e A340526 k   A000041         T(6,k)
%e A340526 1      1  *  14   =   14
%e A340526 2      1  *  10   =   10
%e A340526 3      2  *   8   =   16
%e A340526 4      3  *   5   =   15
%e A340526 5      5  *   3   =   15
%e A340526 6      7  *   1   =    7
%e A340526 .          A006218
%e A340526 --------------------------
%e A340526 The sum of row 6 is 14 + 10 + 16 + 15 + 15 + 7 = 77, equaling A284870(6).
%o A340526 (PARI) f(n) = sum(k=1, n, n\k); \\ A006218
%o A340526 T(n,k) = f(n-k+1)*numbpart(k-1); \\ _Michel Marcus_, Jan 15 2021
%Y A340526 Columns 1 and 2 give A006218.
%Y A340526 Leading diagonal gives A000041.
%Y A340526 Row sums give A284870.
%Y A340526 Cf. A176206, A221531, A339106, A340424, A340425, A340524, A340426, A340527, A336811.
%K A340526 nonn,tabl
%O A340526 1,2
%A A340526 _Omar E. Pol_, Jan 10 2021