A325781 Heinz numbers of complete integer partitions.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 90, 96, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 160, 162, 168, 176, 180, 192, 198, 200, 210, 216, 220, 224, 234, 240, 252, 256, 260, 264, 270, 280, 288, 294, 300

Offset

1,2

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 its divisors have sums of prime indices covering an initial interval of nonnegative integers. For example, the divisors of 60 are {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, with respective sums of prime indices {0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7}, so 60 is in the sequence.

Links

Table of n, a(n) for n=1..58.

Example

The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    24: {1,1,1,2}
    30: {1,2,3}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    42: {1,2,4}
    48: {1,1,1,1,2}
    54: {1,2,2,2}
    56: {1,1,1,4}
    60: {1,1,2,3}
    64: {1,1,1,1,1,1}

Mathematica

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p]*k]];
Select[Range[1000], normQ[hwt/@Rest[Divisors[#]]]&]

Crossrefs

Cf. A002033, A056239, A103295, A112798, A126796, A188431, A299702, A325684, A325770, A325780, A325782.

Sequence in context: A238443 A325795 A359420 * A231565 A191743 A308115

Adjacent sequences: A325778 A325779 A325780 * A325782 A325783 A325784

Keyword

nonn

Author

Gus Wiseman, May 21 2019

Status

approved