A340526 - OEIS

%I #37 Jul 25 2021 13:47:29

%S 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 11,20,16,28,30,40,35,33,15,23,20,32,42,50,56,55,45,22,27,23,40,48,70,

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

%C 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 Conjecture 2: the sum of row n equals A284870, the total number of parts in all partitions of all positive integers <= n.

%C The above conjectures are connected due to the correspondence between divisors and partitions (cf. A336811).

%e Triangle begins:

%e    1;

%e    3,  1;

%e    5,  3,  2;

%e    8,  5,  6,  3;

%e   10,  8, 10,  9,   5;

%e   14, 10, 16, 15,  15,   7;

%e   16, 14, 20, 24,  25,  21,  11;

%e   20, 16, 28, 30,  40,  35,  33,  15;

%e   23, 20, 32, 42,  50,  56,  55,  45,  22;

%e   27, 23, 40, 48,  70,  70,  88,  75,  66,  30;

%e   29, 27, 46, 60,  80,  98, 110, 120, 110,  90,  42;

%e   35, 29, 54, 69, 100, 112, 154, 150, 176, 150, 126,  56;

%e ...

%e For n = 6 the calculation of every term of row 6 is as follows:

%e --------------------------

%e k   A000041         T(6,k)

%e 1      1  *  14   =   14

%e 2      1  *  10   =   10

%e 3      2  *   8   =   16

%e 4      3  *   5   =   15

%e 5      5  *   3   =   15

%e 6      7  *   1   =    7

%e .          A006218

%e --------------------------

%e The sum of row 6 is 14 + 10 + 16 + 15 + 15 + 7 = 77, equaling A284870(6).

%o (PARI) f(n) = sum(k=1, n, n\k); \\ A006218

%o T(n,k) = f(n-k+1)*numbpart(k-1); \\ _Michel Marcus_, Jan 15 2021

%Y Columns 1 and 2 give A006218.

%Y Leading diagonal gives A000041.

%Y Row sums give A284870.

%Y Cf. A176206, A221531, A339106, A340424, A340425, A340524, A340426, A340527, A336811.

%K nonn,tabl

%O 1,2

%A _Omar E. Pol_, Jan 10 2021