A325763 Heinz numbers of integer partitions whose consecutive subsequence-sums cover an initial interval of positive integers.

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 100, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 280, 288, 300, 320, 324, 336, 352, 360, 384, 392, 400, 416, 432, 448, 480, 486, 500, 504, 512

Offset

1,2

Comments

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

The enumeration of these partitions by sum appears to be A002865.

Links

Table of n, a(n) for n=1..57.

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}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    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}
    72: {1,1,1,2,2}
    80: {1,1,1,1,3}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#], {___, s__, ___}:>Plus[s]]]&]

Crossrefs

Cf. A002033, A103295, A103300, A169942, A325676, A325685, A325764, A325765.

Sequence in context: A068563 A124240 A320580 * A363949 A068997 A293928

Adjacent sequences: A325760 A325761 A325762 * A325764 A325765 A325766

Keyword

nonn

Author

Gus Wiseman, May 19 2019

Status

approved