A246867 - OEIS

A246867

Triangle T(n,k) in which n-th row lists in increasing order all partitions lambda of n into distinct parts encoded as Product_{i:lambda} prime(i); n>=0, 1<=k<=A000009(n).

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 13, 21, 22, 30, 17, 26, 33, 35, 42, 19, 34, 39, 55, 66, 70, 23, 38, 51, 65, 77, 78, 105, 110, 29, 46, 57, 85, 91, 102, 130, 154, 165, 210, 31, 58, 69, 95, 114, 119, 143, 170, 182, 195, 231, 330, 37, 62, 87, 115, 133, 138, 187

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OFFSET

0,2

COMMENTS

The concatenation of all rows (with offset 1) gives a permutation of the squarefree numbers A005117. The missing positive numbers are in A013929.

LINKS

Alois P. Heinz, Rows n = 0..42, flattened

EXAMPLE

The partitions of n=5 into distinct parts are {[5], [4,1], [3,2]}, encodings give {prime(5), prime(4)*prime(1), prime(3)*prime(2)} = {11, 7*2, 5*3} => row 5 = [11, 14, 15].

For n=0 the empty partition [] gives the empty product 1.

Triangle T(n,k) begins:

5, 6;

7, 10;

11, 14, 15;

13, 21, 22, 30;

17, 26, 33, 35, 42;

19, 34, 39, 55, 66, 70;

23, 38, 51, 65, 77, 78, 105, 110;

29, 46, 57, 85, 91, 102, 130, 154, 165, 210;

...

Corresponding triangle of strict integer partitions begins:

(1)

(2)

(3) (21)

(4) (31)

(5) (41) (32)

(6) (42) (51) (321)

(7) (61) (52) (43) (421)

(8) (71) (62) (53) (521) (431)

(9) (81) (72) (63) (54) (621) (432) (531). - Gus Wiseman, Feb 23 2018

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [], [seq(

map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))

end:

T:= n-> sort(b(n$2))[]:

seq(T(n), n=0..14);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Flatten[Table[Map[ #*Prime[i]^j&, b[n-i*j, i-1]], {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

CROSSREFS

Column k=1 gives: A008578(n+1).

Last elements of rows give: A246868.

Row sums give A147655.

Row lengths are: A000009.

Cf. A005117, A118462, A215366 (the same for all partitions), A258323, A299755, A299757, A299759.

Sequence in context: A166158 A289997 A137313 * A028805 A322846 A302496

Adjacent sequences: A246864 A246865 A246866 * A246868 A246869 A246870

KEYWORD

nonn,tabf,look

AUTHOR

Alois P. Heinz, Sep 05 2014

STATUS

approved