A378175 - OEIS

%I #28 Nov 20 2024 05:23:22

%S 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 47,55,49,53,57,63,81,59,61,65,69,75,67,71,77,87,99,73,85,79,93,83,89,

%U 91,95,105,111,117,135,97,121,101,123,103,115,125,107,119,129,153

%N 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 Alois P. Heinz, <a href="/A378175/b378175.txt">Rows n = 1..2047, flattened</a>

%F T(prime(n),1) = T(A000040(n),1) = A006450(n).

%e 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 For n=1 the empty partition [] gives the empty product 1.

%e Triangle T(n,k) begins:

%e    1 ;

%e    3 ;

%e    5 ;

%e    7,  9 ;

%e   11 ;

%e   13, 15 ;

%e   17 ;

%e   19, 21, 27 ;

%e   23, 25 ;

%e   29, 33 ;

%e   31 ;

%e   35, 37, 39, 45 ;

%e   41 ;

%e   43, 51 ;

%e   47, 55 ;

%e   49, 53, 57, 63, 81 ;

%e   59 ;

%e   ...

%p b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*

%p       ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})

%p     end:

%p T:= n-> sort([b(n)[]])[]:

%p seq(T(n), n=1..28);

%Y Row sums give A378176.

%Y Row lengths give A001055.

%Y Column k=1 gives A318871.

%Y Rightmost elements of rows give A064988.

%Y Sorted terms give A005408.

%Y Cf. A000040, A006450, A215366, A377852.

%K nonn,tabf

%O 1,2

%A _Alois P. Heinz_, Nov 18 2024