A377852 - OEIS

%I #17 Nov 21 2024 05:33:33

%S 2,3,5,7,9,11,13,30,17,19,84,108,23,200,29,264,31,37,624,1120,1440,41,

%T 43,1632,47,7040,53,3648,12544,16128,20736,59,61,8832,33280,76800,67,

%U 71,22272,157696,202752,73,174080,79,47616,83,89,113664,778240,1490944,1916928,3440640,4423680

%N Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n whose sum is also n (with factors >= 1), encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

%H Alois P. Heinz, <a href="/A377852/b377852.txt">Rows n = 1..1000, flattened</a>

%e The multiplicative partitions of n=8 whose sum is also n are {[8], [4,2,1,1], [2,2,2,1,1]}, encodings give {prime(8), prime(4)*prime(2)*prime(1)^2, prime(2)^3*prime(1)^2} = {19, 7*3*2^2, 3^3*2^2} => row 8 = [19, 84, 108].

%e For n=1 the partition [1] gives prime(1) = 2.

%e Triangle T(n,k) begins:

%e    2 ;

%e    3 ;

%e    5 ;

%e    7,    9 ;

%e   11

%e   13,   30 ;

%e   17 ;

%e   19,   84,   108 ;

%e   23,  200 ;

%e   29,  264 ;

%e   31 ;

%e   37,  624,  1120,  1440 ;

%e   41 ;

%e   43, 1632 ;

%e   47, 7040 ;

%e   53, 3648, 12544, 16128, 20736 ;

%e   59 ;

%e   ...

%Y Column k=1 gives A000040.

%Y Row sums give A377853.

%Y Row lengths give A001055.

%Y Cf. A215366, A378175.

%K nonn,tabf

%O 1,1

%A _Alois P. Heinz_, Nov 09 2024