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A356944
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MM-numbers of multisets of gapless multisets of positive integers. Products of primes indexed by elements of A073491.
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8
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70
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OFFSET
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1,2
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COMMENTS
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A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
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.
We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The size of this multiset of multisets is A302242(n). For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
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LINKS
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EXAMPLE
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The initial terms and corresponding multiset partitions:
1: {}
2: {{}}
3: {{1}}
4: {{},{}}
5: {{2}}
6: {{},{1}}
7: {{1,1}}
8: {{},{},{}}
9: {{1},{1}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Select[Range[100], And@@nogapQ/@primeMS/@primeMS[#]&]
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CROSSREFS
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A000688 counts factorizations into prime powers.
A001222 counts prime factors with multiplicity.
A011782 counts multisets covering an initial interval.
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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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