A258118 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all complete partitions of n.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 30, 36, 40, 48, 64, 42, 54, 56, 60, 72, 80, 96, 128, 84, 90, 100, 108, 112, 120, 144, 160, 192, 256, 126, 132, 140, 150, 162, 168, 176, 180, 200, 216, 224, 240, 288, 320, 384, 512, 198, 210, 220, 252, 264, 270, 280, 300, 324, 336, 352, 360, 400, 432, 448, 480, 576, 640, 768, 1024

Offset: 0

Author

Emeric Deutsch, Jun 07 2015

Keywords

• nonn
• look
• tabf

Comments

A partition of n is complete if every number from 1 to n can be represented as a sum of parts of the partition.

The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,1,4] we get 2*2*2*7 = 56. It is in the sequence because the partition [1,1,1,4] is complete.

Except for a(0)=1, there are no odd numbers in the sequence. Indeed, a partition having an odd Heinz number does not have 1 as a part and, consequently, it cannot be complete.

Number of terms in row n is A126796(n). As a matter of fact, so far, the triangle has been constructed by selecting those A126796(n) entries from row n of A215366 which correspond to complete partitions. Last term in row n is 2^n.

Examples

			54 = 2*3*3*3 is in the sequence because the partition [1,2,2,2] is complete.
28 = 2*2*7 is not in the sequence because the partition [1,1,4] is not complete.
Triangle T(n,k) begins:
   1;
   2;
   4;
   6,  8;
  12, 16;
  18, 20,  24,  32;
  30, 36,  40,  48,  64;
  42, 54,  56,  60,  72,  80,  96, 128;
  84, 90, 100, 108, 112, 120, 144, 160, 192, 256;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..30, flattened
• SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Crossrefs

Cf. A000079, A215366, A126796, A258119.

Column k=1 gives A259941.

Row sums give A360791.

Programs

Maple

T:= proc(m) local b, ll, p;
      p:= proc(l) ll:=ll, (mul(ithprime(j), j=l)); 1 end:
      b:= proc(n, i, l) `if`(i<2, p([l[], 1$n]), `if`(n<2*i-1,
      b(n, iquo(n+1, 2), l), b(n, i-1, l)+b(n-i, i, [l[], i])))
      end: ll:= NULL; b(m, iquo(m+1, 2), []): sort([ll])[]
    end:
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 07 2015

Mathematica

T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, Product[Prime[j], {j, l}]]; 1); b[n_, i_, l_List] := If[i<2, p[Join[l, Array[1&, n]]], If[n < 2*i-1, b[n, Quotient[n+1, 2], l], b[n, i-1, l] + b[n-i, i, Append[l, i] ]]]; ll = {}; b[m, Quotient[m+1, 2], {}]; Sort[ll]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)