A070861 Triangle of all possible distinct numbers obtained as a product of distinct numbers from 1..n.

1, 1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, 36, 40, 48, 60, 72, 90, 120, 144, 180, 240, 360, 720, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42

Offset: 0

Author

Amarnath Murthy, May 16 2002

Keywords

• nonn
• tabf

Comments

Factorials are a subsequence (A000142). - Reinhard Zumkeller, Jul 02 2011

More generally, all sequences of positive integers are subsequences. - Charles R Greathouse IV, Mar 06 2017

Examples

			Triangle begins:
  1;
  1;
  1, 2;
  1, 2, 3, 6;
  1, 2, 3, 4, 6, 8, 12, 24;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..20 of triangle, flattened

Crossrefs

Cf. A000142, A060957.

Row sums give A070863.

Row products give A283261.

Programs

Haskell

a070861 n = a070861_list !! (n-1)
a070861_list = concat a070861_tabf
a070861_tabf = [1] : f 2 [1] where
   f n ps = ps' : f (n+1) ps' where ps' = m ps $ map (n*) ps
   m []         ys = ys
   m xs'@(x:xs) ys'@(y:ys)
       | x < y     = x : m xs ys'
       | x == y    = x : m xs ys
       | otherwise = y : m xs' ys
b070861 = bFile' "A070861" (concat $ take 20 a070861_tabf) 1
-- Reinhard Zumkeller, Jul 02 2011

Maple

T:= proc(n) option remember; `if`(n=0, 1,
      sort([map(x-> [x, x*n][], {T(n-1)})[]])[])
    end:
seq(T(n), n=0..7);  # Alois P. Heinz, Aug 01 2022

Mathematica

row[n_] := Times @@@ Subsets[Range[n]] // Flatten // Union; Table[row[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 02 2015 *)

PARI

row(n)=my(v=[2..n]); Set(vector(2^(n-1),i, factorback(vecextract(v,i-1)))) \\ Charles R Greathouse IV, Mar 06 2017

Formula

T(n,A060957(n)) = A000142(n) = n!. - Alois P. Heinz, Aug 01 2022

Extensions

Corrected and extended by Lior Manor, May 23 2002

Row n=0 prepended by Alois P. Heinz, Aug 01 2022