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A353867
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Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.
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19
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1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936
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OFFSET
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1,2
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COMMENTS
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The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Related concepts:
- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).
- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.
- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).
- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
16: {1,1,1,1}
20: {1,1,3}
30: {1,2,3}
32: {1,1,1,1,1}
56: {1,1,1,4}
64: {1,1,1,1,1,1}
90: {1,2,2,3}
128: {1,1,1,1,1,1,1}
140: {1,1,3,4}
176: {1,1,1,1,5}
210: {1,2,3,4}
256: {1,1,1,1,1,1,1,1}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
norqQ[m_]:=Sort[m]==Range[0, Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t, i], {t, Split[s]}, {i, 0, Length[t]}]];
Select[Range[1000], norqQ[Total/@Select[msubs[primeMS[#]], SameQ@@#&]]&]
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CROSSREFS
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These partitions are counted by A353865.
A073093 counts prime-power divisors.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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