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 A329552 Smallest MM-number of a connected set of n sets. 7
 1, 2, 39, 195, 5655, 62205, 2674815 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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}}. LINKS EXAMPLE 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}} MATHEMATICA 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[]], LCM@@s[[c[]]]]]]]]; da=Select[Range, 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&]]}] CROSSREFS MM-numbers of connected set-systems are A328514. The weight of the system with MM-number n is A302242(n). Connected numbers are A305078. 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. Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters). Sequence in context: A175760 A080920 A086338 * A042683 A319520 A209632 Adjacent sequences:  A329549 A329550 A329551 * A329553 A329554 A329555 KEYWORD nonn,more AUTHOR Gus Wiseman, Nov 17 2019 STATUS approved

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Last modified July 24 21:47 EDT 2021. Contains 346273 sequences. (Running on oeis4.)