A258119 Triangle T(n,k) in which the n-th row lists in increasing order the Heinz numbers of all perfect partitions of n.

1, 2, 4, 6, 8, 16, 18, 20, 32, 64, 42, 54, 56, 128, 100, 256, 162, 176, 512, 1024, 234, 260, 294, 392, 416, 486, 500, 2048, 4096, 1088, 1458, 8192, 1936, 2500, 16384, 798, 1026, 1064, 2058, 2432, 2744, 4374, 32768, 65536, 2300, 3042, 3380, 5408, 5888, 12500, 13122, 131072

Offset: 0

Author

Emeric Deutsch, Jun 07 2015

Keywords

• nonn
• look
• tabf

Comments

A partition of n is perfect if it contains just one partition of every number less than n when repeated parts are regarded as indistinguishable.

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 perfect.

Number of terms in row n is A002033(n). As a matter of fact, so far the triangle has been constructed by selecting those A002033(n) entries from row n of A215366 which correspond to perfect 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 perfect.
24 = 2*2*2*3 is not in the sequence because the partition [1,1,1,2] is not perfect (1+1+1=1+2; it is complete).
Triangle T(n,k) begins:
   1;
   2;
   4;
   6,  8;
  16;
  18, 20, 32;
  64;
  42, 54, 56, 128;
  ...
		

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p 123.

Links

• Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Cf. A000079, A215366, A002033, A258118.

Column k=1 gives A259939.

Row sums give A360713.

Programs

Maple

with(numtheory):
T:= proc(m) local b, ll, p;
      if m=0 then return 1 fi;
      p:= proc(l) ll:=ll, 2^(l[1]-1)*mul(ithprime(
      mul(l[j], j=1..i-1))^(l[i]-1), i=2..nops(l)) end:
      b:= proc(n, l) `if`(n=1, p(l), seq(b(n/d, [l[], d])
      , d=divisors(n) minus{1})) end:
      ll:= NULL; b(m+1, []): sort([ll])[]
    end:
seq(T(n), n=0..20);  # Alois P. Heinz, Jun 08 2015

Mathematica

T[0] = {1}; T[m_] := Module[{b, ll, p}, p[l_List] := (ll = Append[ll, 2^(l[[1]]-1)*Product[Prime[Product[l[[j]], {j, 1, i-1}]]^(l[[i]]-1), {i, 2, Length[l]}]]; 1); b[n_, l_List] := If[n == 1, p[l], Table[b[n/d, Append[l, d]], {d, Divisors[n] // Rest}]]; ll = {}; b[m+1, {}]; Sort[ll] ]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 28 2016, after Alois P. Heinz *)