A215366 Triangle T(n,k) read by rows in which n-th row lists in increasing order all partitions lambda of n encoded as Product_{i in lambda} prime(i); n>=0, 1<=k<=A000041(n).

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128, 19, 34, 39, 49, 52, 55, 63, 66, 70, 75, 81, 84, 88, 90, 100, 108, 112, 120, 144, 160, 192, 256

Offset

0,2

Comments

The concatenation of all rows (with offset 1) gives a permutation of the natural numbers A000027 with fixed points 1-6, 9, 10, 14, 15, 21, 22, 33, 49, 1095199, ... and inverse permutation A215501.

Number m is positioned in row n = A056239(m). The number of different values m, such that both m and m+1 occur in row n is A088850(n). A215369 lists all values m, such that both m and m+1 are in the same row.

The power prime(i)^j of the i-th prime is in row i*j for j in {0,1,2, ... }.

Column k=2 contains the even semiprimes A100484, where 10 and 22 are replaced by the odd semiprimes 9 and 21, respectively.

This triangle is related to the triangle A145518, see in both triangles the first column, the right border, the second right border and the row sums. - Omar E. Pol, May 18 2015

Links

Alois P. Heinz, Rows n = 0..26, flattened

Index entries for sequences that are permutations of the natural numbers

Index entries for sequences computed from indices in prime factorization

Formula

Recurrence relation, explained for the set S(4) of entries in row 4: multiply the entries of S(3) by 2 (= 1st prime), multiply the entries of S(2) by 3 (= 2nd prime), multiply the entries of S(1) by 5 (= 3rd prime), multiply the entries of S(0) by 7 (= 4th prime); take the union of all the obtained products. The 3rd Maple program is based on this recurrence relation. - Emeric Deutsch, Jan 23 2016

Example

The partitions of n=3 are {[3], [2,1], [1,1,1]}, encodings give {prime(3), prime(2)*prime(1), prime(1)^3} = {5, 3*2, 2^3} => row 3 = [5, 6, 8].
For n=0 the empty partition [] gives the empty product 1.
Triangle T(n,k) begins:
   1;
   2;
   3,  4;
   5,  6,  8;
   7,  9, 10, 12, 16;
  11, 14, 15, 18, 20, 24, 32;
  13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64;
  17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128;
  ...
Corresponding triangle of integer partitions begins:
  ();
  1;
  2, 11;
  3, 21, 111;
  4, 22, 31, 211, 1111;
  5, 41, 32, 221, 311, 2111, 11111;
  6, 42, 51, 33, 222, 411, 321, 2211, 3111, 21111, 111111;
  7, 61, 52, 43, 421, 511, 322, 331, 2221, 4111, 3211, 22111, 31111, 211111, 1111111;  - Gus Wiseman, Dec 12 2016

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
       [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
T:= n-> sort(b(n, n))[]:
seq(T(n), n=0..10);
# (2nd Maple program)
with(combinat): A := proc (n) local P, A, i: P := partition(n): A := {}; for i to nops(P) do A := `union`(A, {mul(ithprime(P[i][j]), j = 1 .. nops(P[i]))}) end do: A end proc; # the command A(m) yields row m. # Emeric Deutsch, Jan 23 2016
# (3rd Maple program)
q:= 7: S[0] := {1}: for m to q do S[m] := `union`(seq(map(proc (f) options operator, arrow: ithprime(j)*f end proc, S[m-j]), j = 1 .. m)) end do; # for a given positive integer q, the program yields rows 0, 1, 2, ..., q. # Emeric Deutsch, Jan 23 2016

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i<2, {2^n}, Table[Function[#*Prime[i]^j] /@ b[n - i*j, i-1], {j, 0, n/i}] // Flatten]; T[n_] := Sort[b[n, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)
nn=7; HeinzPartition[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]//Reverse];
Take[GatherBy[Range[2^nn], Composition[Total, HeinzPartition]], nn+1] (* Gus Wiseman, Dec 12 2016 *)
Table[Map[Times @@ Prime@ # &, IntegerPartitions[n]], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

Prog

(PARI) \\ From M. F. Hasler, Dec 06 2016 (Start)
A215366_row(n)=vecsort([vecprod([prime(p)|p<-P])|P<-partitions(n)]) \\ bug fix & syntax update by M. F. Hasler, Oct 20 2023
A215366_vec(N)=concat(apply(A215366_row, [0..N])) \\ "flattened" rows 0..N (End)

Crossrefs

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

Last elements of rows give: A000079.

Second to last elements of rows give: A007283(n-2) for n>1.

Row sums give: A145519.

Row lengths are: A000041.

Cf. A129129 (with row elements using order of A080577).

LCM of terms in row n gives A138534(n).

Cf. A000027, A000040, A056239, A063008, A088850, A100484, A215501.

Cf. A112798, A246867 (the same for partitions into distinct parts).

Cf. A324939.

Sequence in context: A358122 A334111 A243571 * A333483 A334433 A334435

Adjacent sequences: A215363 A215364 A215365 * A215367 A215368 A215369

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Aug 08 2012

Status

approved