A070861 - OEIS

%I #28 Aug 01 2022 12:40:59

%S 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,

%T 60,120,1,2,3,4,5,6,8,10,12,15,18,20,24,30,36,40,48,60,72,90,120,144,

%U 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

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

%C Factorials are a subsequence (A000142). - _Reinhard Zumkeller_, Jul 02 2011

%C More generally, all sequences of positive integers are subsequences. - _Charles R Greathouse IV_, Mar 06 2017

%H Reinhard Zumkeller, <a href="/A070861/b070861.txt">Rows n = 0..20 of triangle, flattened</a>

%F T(n,A060957(n)) = A000142(n) = n!. - _Alois P. Heinz_, Aug 01 2022

%e Triangle begins:

%e   1;

%e   1;

%e   1, 2;

%e   1, 2, 3, 6;

%e   1, 2, 3, 4, 6, 8, 12, 24;

%e   ...

%p T:= proc(n) option remember; `if`(n=0, 1,

%p       sort([map(x-> [x, x*n][], {T(n-1)})[]])[])

%p     end:

%p seq(T(n), n=0..7);  # _Alois P. Heinz_, Aug 01 2022

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

%o (Haskell)

%o a070861 n = a070861_list !! (n-1)

%o a070861_list = concat a070861_tabf

%o a070861_tabf = [1] : f 2 [1] where

%o    f n ps = ps' : f (n+1) ps' where ps' = m ps $ map (n*) ps

%o    m []         ys = ys

%o    m xs'@(x:xs) ys'@(y:ys)

%o        | x < y     = x : m xs ys'

%o        | x == y    = x : m xs ys

%o        | otherwise = y : m xs' ys

%o b070861 = bFile' "A070861" (concat $ take 20 a070861_tabf) 1

%o -- _Reinhard Zumkeller_, Jul 02 2011

%o (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

%Y Cf. A000142, A060957.

%Y Row sums give A070863.

%Y Row products give A283261.

%K nonn,tabf

%O 0,4

%A _Amarnath Murthy_, May 16 2002

%E Corrected and extended by _Lior Manor_ May 23 2002

%E Row n=0 prepended by _Alois P. Heinz_, Aug 01 2022