A325764 Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.

1, 2, 4, 6, 8, 16, 18, 20, 32, 54, 56, 64, 100, 128, 162, 176, 256, 392, 416, 486, 500, 512, 1024, 1088, 1458, 1936, 2048, 2432, 2500, 2744, 4096, 4374, 5408, 5888, 8192, 12500, 13122, 14848, 16384, 18496, 19208, 21296, 31744, 32768, 39366, 46208, 62500, 65536

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 is given by A325765.

Links

Table of n, a(n) for n=1..48.

Example

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}
    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}
   256: {1,1,1,1,1,1,1,1}
   392: {1,1,1,4,4}
   416: {1,1,1,1,1,6}
   486: {1,2,2,2,2,2}
   500: {1,1,3,3,3}
   512: {1,1,1,1,1,1,1,1,1}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], UnsameQ@@Total/@Union[ReplaceList[primeMS[#], {___, s__, ___}:>{s}]]&&Range[Total[primeMS[#]]]==Union[ReplaceList[primeMS[#], {___, s__, ___}:>Plus[s]]]&]

Crossrefs

Cf. A002033, A056239, A103295, A103300, A112798, A143823, A169942, A325676, A325685, A325763, A325765, A325769, A325770.

Sequence in context: A325780 A283423 A073935 * A258119 A073696 A058602

Adjacent sequences: A325761 A325762 A325763 * A325765 A325766 A325767

Keyword

nonn

Author

Gus Wiseman, May 20 2019

Status

approved