

A329553


Smallest MMnumber of a connected set of n multisets.


3




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 MMnumber 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 MMnumber 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

MMnumbers of connected sets of sets are A328514.
The weight of the system with MMnumber n is A302242(n).
Maximum connected divisor is A327076.
BIInumbers of connected setsystems are A326749.
The smallest BIInumber of a connected setsystem is A329625.
The case of strict edges is A329552.
The smallest MMnumber of a set of n nonempty sets is A329557(n).


KEYWORD

nonn,more


AUTHOR



STATUS

approved



