A265146 Triangle T(n,k) in which n-th row lists the parts i_1<i_2<...<i_m of the unique strict partition with encoding n = Product_{j=1..m} prime(i_j-j+1); n>=1, 1<=k<=A001222(n).

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 2, 3, 1, 4, 5, 1, 2, 4, 6, 1, 5, 2, 4, 1, 2, 3, 4, 7, 1, 3, 4, 8, 1, 2, 5, 2, 5, 1, 6, 9, 1, 2, 3, 5, 3, 4, 1, 7, 2, 3, 4, 1, 2, 6, 10, 1, 3, 5, 11, 1, 2, 3, 4, 5, 2, 6, 1, 8, 3, 5, 1, 2, 4, 5, 12, 1, 9, 2, 7, 1, 2, 3, 6, 13, 1

Offset

1,2

Comments

A strict partition is a partition into distinct parts.

Row n=1 contains the parts of the empty partition, so it is empty.

Links

Alois P. Heinz, Rows n = 1..1000, flattened

Formula

T(prime(n),1) = n.

Example

n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].
Triangle T(n,k) begins:
01 :  ;
02 :  1;
03 :  2;
04 :  1, 2;
05 :  3;
06 :  1, 3;
07 :  4;
08 :  1, 2, 3;
09 :  2, 3;
10 :  1, 4;
11 :  5;
12 :  1, 2, 4;
13 :  6;
14 :  1, 5;
15 :  2, 4;
16 :  1, 2, 3, 4;

Maple

T:= n-> ((l-> seq(l[j]+j-1, j=1..nops(l)))(sort([seq(
       numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(T(n), n=1..100);

Mathematica

T[n_] := Function[l, Table[l[[j]]+j-1, {j, 1, Length[l]}]][Sort[ Flatten[ Table[ Array[ PrimePi[i[[1]]]&, i[[2]]], {i, FactorInteger[n]}]]]];
Table[T[n], {n, 1, 100}] // Flatten // Rest (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Crossrefs

Column k=1 gives A055396 (for n>1).

Last terms of rows give A252464 (for n>1).

Row sums give A266475.

Cf. A000009, A000040, A001222, A112798, A246688, A265145.

Sequence in context: A036995 A225597 A116908 * A358136 A325537 A072851

Adjacent sequences: A265143 A265144 A265145 * A265147 A265148 A265149

Keyword

nonn,tabf

Author

Alois P. Heinz, Dec 02 2015

Status

approved