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

0,4

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

Links

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

Formula

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000142, A060957.

Row sums give A070863.

Row products give A283261.

Sequence in context: A078777 A135938 A079210 * A277566 A261144 A106524

Adjacent sequences: A070858 A070859 A070860 * A070862 A070863 A070864

Keyword

nonn,tabf

Author

Amarnath Murthy, May 16 2002

Extensions

Corrected and extended by Lior Manor May 23 2002

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

Status

approved