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A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).

%I A378175 #28 Nov 20 2024 05:23:22
%S A378175 1,3,5,7,9,11,13,15,17,19,21,27,23,25,29,33,31,35,37,39,45,41,43,51,
%T A378175 47,55,49,53,57,63,81,59,61,65,69,75,67,71,77,87,99,73,85,79,93,83,89,
%U A378175 91,95,105,111,117,135,97,121,101,123,103,115,125,107,119,129,153
%N A378175 Triangle T(n,k) read by rows in which n-th row lists in increasing order all multiplicative partitions mu of n (with factors > 1) encoded as Product_{j in mu} prime(j); n>=1, 1<=k<=A001055(n).
%H A378175 Alois P. Heinz, <a href="/A378175/b378175.txt">Rows n = 1..2047, flattened</a>
%F A378175 T(prime(n),1) = T(A000040(n),1) = A006450(n).
%e A378175 The multiplicative partitions of n=8 are {[8], [4,2], [2,2,2]}, encodings give {prime(8), prime(4)*prime(2), prime(2)^3} = {19, 7*3, 3^3} => row 8 = [19, 21, 27].
%e A378175 For n=1 the empty partition [] gives the empty product 1.
%e A378175 Triangle T(n,k) begins:
%e A378175    1 ;
%e A378175    3 ;
%e A378175    5 ;
%e A378175    7,  9 ;
%e A378175   11 ;
%e A378175   13, 15 ;
%e A378175   17 ;
%e A378175   19, 21, 27 ;
%e A378175   23, 25 ;
%e A378175   29, 33 ;
%e A378175   31 ;
%e A378175   35, 37, 39, 45 ;
%e A378175   41 ;
%e A378175   43, 51 ;
%e A378175   47, 55 ;
%e A378175   49, 53, 57, 63, 81 ;
%e A378175   59 ;
%e A378175   ...
%p A378175 b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
%p A378175       ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
%p A378175     end:
%p A378175 T:= n-> sort([b(n)[]])[]:
%p A378175 seq(T(n), n=1..28);
%Y A378175 Row sums give A378176.
%Y A378175 Row lengths give A001055.
%Y A378175 Column k=1 gives A318871.
%Y A378175 Rightmost elements of rows give A064988.
%Y A378175 Sorted terms give A005408.
%Y A378175 Cf. A000040, A006450, A215366, A377852.
%K A378175 nonn,tabf
%O A378175 1,2
%A A378175 _Alois P. Heinz_, Nov 18 2024