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

0,2

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.

References

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

Links

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

Example

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;
  ...

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 *)

Crossrefs

Cf. A000079, A215366, A002033, A258118.

Column k=1 gives A259939.

Row sums give A360713.

Sequence in context: A283423 A073935 A325764 * A073696 A058602 A133808

Adjacent sequences: A258116 A258117 A258118 * A258120 A258121 A258122

Keyword

nonn,look,tabf

Author

Emeric Deutsch, Jun 07 2015

Status

approved