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A325764
Heinz numbers of integer partitions whose distinct consecutive subsequences have distinct sums that cover an initial interval of positive integers.
2
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.
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]]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 20 2019
STATUS
approved