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).

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 27, 23, 25, 29, 33, 31, 35, 37, 39, 45, 41, 43, 51, 47, 55, 49, 53, 57, 63, 81, 59, 61, 65, 69, 75, 67, 71, 77, 87, 99, 73, 85, 79, 93, 83, 89, 91, 95, 105, 111, 117, 135, 97, 121, 101, 123, 103, 115, 125, 107, 119, 129, 153

Offset: 1

Author

Alois P. Heinz, Nov 18 2024

Keywords

• nonn
• tabf

Examples

			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].
For n=1 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1 ;
   3 ;
   5 ;
   7,  9 ;
  11 ;
  13, 15 ;
  17 ;
  19, 21, 27 ;
  23, 25 ;
  29, 33 ;
  31 ;
  35, 37, 39, 45 ;
  41 ;
  43, 51 ;
  47, 55 ;
  49, 53, 57, 63, 81 ;
  59 ;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..2047, flattened

Crossrefs

Row sums give A378176.

Row lengths give A001055.

Column k=1 gives A318871.

Rightmost elements of rows give A064988.

Sorted terms give A005408.

Cf. A000040, A006450, A215366, A377852.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, {1}, {seq(map(x-> x*
      ithprime(d), b(n/d))[], d=numtheory[divisors](n) minus {1})})
    end:
T:= n-> sort([b(n)[]])[]:
seq(T(n), n=1..28);

Formula

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