

A329552


Smallest MMnumber of a connected set of n sets.


7




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

Table of n, a(n) for n=0..6.


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[[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&]]}]


CROSSREFS

MMnumbers of connected setsystems are A328514.
The weight of the system with MMnumber n is A302242(n).
Connected numbers are A305078.
Maximum connected divisor is A327076.
BIInumbers of connected sets of sets are A326749.
The smallest BIInumber 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 MMnumber 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 MMnumbers: 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



