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A354583
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Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.
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3
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12, 24, 36, 40, 48, 60, 63, 72, 80, 84, 96, 108, 112, 120, 126, 132, 144, 156, 160, 168, 180, 189, 192, 200, 204, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 320, 324, 325, 336, 348, 351, 352, 360, 372, 378, 384, 396, 400, 408, 420, 432, 440
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OFFSET
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1,1
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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.
The term rucksack is short for run-knapsack.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
12: {1,1,2}
24: {1,1,1,2}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
60: {1,1,2,3}
63: {2,2,4}
72: {1,1,1,2,2}
80: {1,1,1,1,3}
84: {1,1,2,4}
96: {1,1,1,1,1,2}
108: {1,1,2,2,2}
112: {1,1,1,1,4}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
144: {1,1,1,1,2,2}
156: {1,1,2,6}
160: {1,1,1,1,1,3}
168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !UnsameQ@@Total/@primeMS/@Select[Divisors[#], PrimePowerQ]&]
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CROSSREFS
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Non-knapsack partitions are ranked by A299729.
The complement for compositions is counted by A354580.
A073093 counts prime-power divisors.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353861 counts distinct partial run-sums of prime indices.
Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, 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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