A325780 Heinz numbers of perfect integer partitions.

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Author

Gus Wiseman, May 21 2019

Keywords

• nonn

Comments

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

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}
		

Crossrefs

Equals the sorted concatenation of the triangle A258119.

A subsequence of A299702 and A325781.

Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Formula

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).