A326037 Heinz numbers of uniform perfect integer partitions.

1, 2, 4, 6, 8, 16, 32, 42, 64, 100, 128, 256, 512, 798, 1024, 2048, 2744, 4096, 8192, 16384, 32768, 42294, 52900, 65536

Offset

1,2

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

An integer partition of n is uniform if all parts appear with the same multiplicity, and perfect if every nonnegative integer up to n is the sum of a unique submultiset.

The enumeration of these partitions by sum is given by A089723.

Links

Table of n, a(n) for n=1..24.

Formula

Intersection of A072774 (uniform), A299702 (knapsack), and A325781 (complete).

Example

The sequence of all uniform perfect integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (11)
      6: (21)
      8: (111)
     16: (1111)
     32: (11111)
     42: (421)
     64: (111111)
    100: (3311)
    128: (1111111)
    256: (11111111)
    512: (111111111)
    798: (8421)
   1024: (1111111111)
   2048: (11111111111)
   2744: (444111)
   4096: (111111111111)
   8192: (1111111111111)
  16384: (11111111111111)

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], SameQ@@Last/@FactorInteger[#]&&Sort[hwt/@Divisors[#]]==Range[0, hwt[#]]&]

Crossrefs

Cf. A002033, A047966, A070941, A072774, A108917, A126796, A276024, A299702.

Cf. A325780, A325781, A326020, A326035, A326036.

Sequence in context: A324579 A271520 A122408 * A100055 A334031 A304660

Adjacent sequences: A326034 A326035 A326036 * A326038 A326039 A326040

Keyword

nonn,more

Author

Gus Wiseman, Jun 04 2019

Status

approved