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

Author

Gus Wiseman, Jun 04 2019

Keywords

• nonn
• more

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.

Examples

			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)
		

Crossrefs

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

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

Programs

Mathematica

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

Formula

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