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

%I A265146 #24 Mar 23 2017 04:15:36
%S A265146 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 A265146 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 A265146 1,8,3,5,1,2,4,5,12,1,9,2,7,1,2,3,6,13,1
%N 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).
%C A265146 A strict partition is a partition into distinct parts.
%C A265146 Row n=1 contains the parts of the empty partition, so it is empty.
%H A265146 Alois P. Heinz, <a href="/A265146/b265146.txt">Rows n = 1..1000, flattened</a>
%F A265146 T(prime(n),1) = n.
%e A265146 n = 12 = 2*2*3 = prime(1)*prime(1)*prime(2) encodes strict partition [1,2,4].
%e A265146 Triangle T(n,k) begins:
%e A265146 01 :  ;
%e A265146 02 :  1;
%e A265146 03 :  2;
%e A265146 04 :  1, 2;
%e A265146 05 :  3;
%e A265146 06 :  1, 3;
%e A265146 07 :  4;
%e A265146 08 :  1, 2, 3;
%e A265146 09 :  2, 3;
%e A265146 10 :  1, 4;
%e A265146 11 :  5;
%e A265146 12 :  1, 2, 4;
%e A265146 13 :  6;
%e A265146 14 :  1, 5;
%e A265146 15 :  2, 4;
%e A265146 16 :  1, 2, 3, 4;
%p A265146 T:= n-> ((l-> seq(l[j]+j-1, j=1..nops(l)))(sort([seq(
%p A265146        numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
%p A265146 seq(T(n), n=1..100);
%t A265146 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 A265146 Table[T[n], {n, 1, 100}] // Flatten // Rest (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)
%Y A265146 Column k=1 gives A055396 (for n>1).
%Y A265146 Last terms of rows give A252464 (for n>1).
%Y A265146 Row sums give A266475.
%Y A265146 Cf. A000009, A000040, A001222, A112798, A246688, A265145.
%K A265146 nonn,tabf
%O A265146 1,2
%A A265146 _Alois P. Heinz_, Dec 02 2015