%I #24 Mar 23 2017 04:15:36

%S 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,

%T 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,

%U 1,8,3,5,1,2,4,5,12,1,9,2,7,1,2,3,6,13,1

%N 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).

%C A strict partition is a partition into distinct parts.

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

%H Alois P. Heinz, <a href="/A265146/b265146.txt">Rows n = 1..1000, flattened</a>

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

%e n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].

%e Triangle T(n,k) begins:

%e 01 : ;

%e 02 : 1;

%e 03 : 2;

%e 04 : 1, 2;

%e 05 : 3;

%e 06 : 1, 3;

%e 07 : 4;

%e 08 : 1, 2, 3;

%e 09 : 2, 3;

%e 10 : 1, 4;

%e 11 : 5;

%e 12 : 1, 2, 4;

%e 13 : 6;

%e 14 : 1, 5;

%e 15 : 2, 4;

%e 16 : 1, 2, 3, 4;

%p T:= n-> ((l-> seq(l[j]+j-1, j=1..nops(l)))(sort([seq(

%p numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):

%p seq(T(n), n=1..100);

%t 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]}]]]];

%t Table[T[n], {n, 1, 100}] // Flatten // Rest (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)

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

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

%Y Row sums give A266475.

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

%K nonn,tabf

%O 1,2

%A _Alois P. Heinz_, Dec 02 2015