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A377852 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).

%I A377852 #17 Nov 21 2024 05:33:33
%S A377852 2,3,5,7,9,11,13,30,17,19,84,108,23,200,29,264,31,37,624,1120,1440,41,
%T A377852 43,1632,47,7040,53,3648,12544,16128,20736,59,61,8832,33280,76800,67,
%U A377852 71,22272,157696,202752,73,174080,79,47616,83,89,113664,778240,1490944,1916928,3440640,4423680
%N A377852 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 A377852 Alois P. Heinz, <a href="/A377852/b377852.txt">Rows n = 1..1000, flattened</a>
%e A377852 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 A377852 For n=1 the partition [1] gives prime(1) = 2.
%e A377852 Triangle T(n,k) begins:
%e A377852    2 ;
%e A377852    3 ;
%e A377852    5 ;
%e A377852    7,    9 ;
%e A377852   11
%e A377852   13,   30 ;
%e A377852   17 ;
%e A377852   19,   84,   108 ;
%e A377852   23,  200 ;
%e A377852   29,  264 ;
%e A377852   31 ;
%e A377852   37,  624,  1120,  1440 ;
%e A377852   41 ;
%e A377852   43, 1632 ;
%e A377852   47, 7040 ;
%e A377852   53, 3648, 12544, 16128, 20736 ;
%e A377852   59 ;
%e A377852   ...
%Y A377852 Column k=1 gives A000040.
%Y A377852 Row sums give A377853.
%Y A377852 Row lengths give A001055.
%Y A377852 Cf. A215366, A378175.
%K A377852 nonn,tabf
%O A377852 1,1
%A A377852 _Alois P. Heinz_, Nov 09 2024