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A329553 Smallest MM-number of a connected set of n multisets. 3
1, 2, 21, 195, 1365, 25935, 435435 (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: {{}}
21: {{1},{1,1}}
195: {{1},{2},{1,2}}
1365: {{1},{2},{1,1},{1,2}}
25935: {{1},{2},{1,1},{1,2},{1,1,1}}
435435: {{1},{2},{1,1},{3},{1,2},{1,3}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], GCD@@s[[#]]>1&]}, If[c=={}, s, zsm[Sort[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000], SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[dae[[Position[PrimeOmega/@dae, k][[1, 1]]]], {k, First[Split[Union[PrimeOmega/@dae], #2==#1+1&]]}]
CROSSREFS
MM-numbers of connected sets of sets 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 set-systems are A326749.
The smallest BII-number of a connected set-system is A329625.
The case of strict edges is A329552.
The smallest MM-number of a set of n nonempty sets is A329557(n).
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).
Sequence in context: A037630 A037756 A037644 * A365061 A110253 A185634
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 17 2019
STATUS
approved

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Last modified February 29 01:59 EST 2024. Contains 370401 sequences. (Running on oeis4.)