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A329552
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Smallest MM-number of a connected set of n sets.
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7
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OFFSET
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0,2
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COMMENTS
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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 of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of 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 sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
39: {{1},{1,2}}
195: {{1},{2},{1,2}}
5655: {{1},{2},{1,2},{1,3}}
62205: {{1},{2},{3},{1,2},{1,3}}
2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
da=Select[Range[10000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da, n][[1, 1]]]], {n, First[Split[Union[PrimeOmega/@da], #2==#1+1&]]}]
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CROSSREFS
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MM-numbers of connected set-systems are A328514.
The weight of the system with MM-number n is A302242(n).
Maximum connected divisor is A327076.
BII-numbers of connected sets of sets are A326749.
The smallest BII-number of a connected set of n sets is A329625(n).
Allowing edges to have repeated vertices gives A329553.
Requiring the edges to form an antichain gives A329555.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Cf. A048143, A056239, A112798, A302494, A304714, A304716, A305079, A322389, A328513, A329554, A329556, A329558.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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