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A212721 Triangle read by rows: n-th row gives distinct products of partitions of n (A000041). 6
1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A034891(n) = length of n-th row;

A000792(n) = largest term of n-th row;

for n>5: A007918(n) = smallest number <= A000792(n) not occurring in n-th row.

LINKS

Reinhard Zumkeller, Rows n = 0..36 of triangle, flattened

EXAMPLE

A000041(6)=11, the 11 partitions and their products of 6:

   1: (1,1,1,1,1,1)   ->   1 * 1 * 1 * 1 * 1 * 1 = 1

   2: (1,1,1,1,2)     ->   1 * 1 * 1 * 1 * 2     = 2

   3: (1,1,1,3)       ->   1 * 1 * 1 * 3         = 3

   4: (1,1,2,2)       ->   1 * 1 * 2 * 2         = 4

   5: (1,1,4)         ->   1 * 1 * 4             = 4

   6: (1,2,3)         ->   1 * 2 * 3             = 6

   7: (1,5)           ->   1 * 5                 = 5

   8: (2,2,2)         ->   2 * 2 * 2             = 8

   9: (2,4)           ->   2 * 4                 = 8

  10: (3,3)           ->   3 * 3                 = 9

  11: (6)             ->                           6,

sorted and duplicates removed: T(6,1..8)=[1,2,3,4,5,6,8,9], A034891(6)=8.

The triangle begins:

   0 |  [1]

   1 |  [1]

   2 |  [1,2]

   3 |  [1,2,3]

   4 |  [1,2,3,4]

   5 |  [1,2,3,4,5,6]

   6 |  [1,2,3,4,5,6,8,9]

   7 |  [1,2,3,4,5,6,7,8,9,10,12]

   8 |  [1,2,3,4,5,6,7,8,9,10,12,15,16,18]

   9 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,24,27]

  10 |  [1,2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21,24,25,27,30,32,36].

MATHEMATICA

row[n_] := Union[Times @@@ IntegerPartitions[n]];

Table[row[n], {n, 0, 10}] (* Jean-Fran├žois Alcover, Jun 29 2019 *)

PROG

(Haskell)

import Data.List (nub, sort)

a212721 n k = a212721_row n !! (k-1)

a212721_row = nub . sort . (map product) . ps 1 where

   ps x 0 = [[]]

   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]

a212721_tabf = map a212721_row [0..]

(Sage)

[sorted(list(set([mul(p) for p in Partitions(n)]))) for n in range(11)] # Peter Luschny, Dec 13 2015

CROSSREFS

Cf. A000041, A000792, A034891.

Sequence in context: A243712 A256553 A194896 * A222417 A253573 A229945

Adjacent sequences:  A212718 A212719 A212720 * A212722 A212723 A212724

KEYWORD

nonn,tabf,look

AUTHOR

Reinhard Zumkeller, Jun 14 2012

STATUS

approved

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Last modified May 13 04:09 EDT 2021. Contains 343836 sequences. (Running on oeis4.)