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A326255
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MM-numbers of capturing multiset partitions.
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15
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667, 989, 1334, 1633, 1769, 1817, 1978, 2001, 2021, 2323, 2461, 2599, 2623, 2668, 2967, 2987, 3197, 3266, 3335, 3538, 3634, 3713, 3749, 3956, 3979, 4002, 4042, 4163, 4171, 4331, 4379, 4429, 4439, 4577, 4646, 4669, 4747, 4819, 4859, 4899, 4922, 4945, 5029, 5198
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OFFSET
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1,1
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COMMENTS
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First differs from A326256 in having 2599.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.
A multiset partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and t < y or z < x and y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.
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LINKS
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EXAMPLE
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The sequence of terms together with their multiset multisystems begins:
667: {{2,2},{1,3}}
989: {{2,2},{1,4}}
1334: {{},{2,2},{1,3}}
1633: {{2,2},{1,1,3}}
1769: {{1,3},{1,2,2}}
1817: {{2,2},{1,5}}
1978: {{},{2,2},{1,4}}
2001: {{1},{2,2},{1,3}}
2021: {{1,4},{2,3}}
2323: {{2,2},{1,6}}
2461: {{2,2},{1,1,4}}
2599: {{2,2},{1,2,3}}
2623: {{1,4},{1,2,2}}
2668: {{},{},{2,2},{1,3}}
2967: {{1},{2,2},{1,4}}
2987: {{1,3},{2,2,2}}
3197: {{2,2},{1,7}}
3266: {{},{2,2},{1,1,3}}
3335: {{2},{2,2},{1,3}}
3538: {{},{1,3},{1,2,2}}
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MATHEMATICA
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capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z&&y>t||x>z&&y<t];
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], capXQ[primeMS/@primeMS[#]]&]
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CROSSREFS
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MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of crossing capturing multiset partitions are A326259.
Capturing set partitions are A326243.
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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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