A329556 - OEIS

A329556

Smallest MM-number of a set of n sets with no singletons.

1, 2, 26, 754, 32422, 1523834

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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

Table of n, a(n) for n=0..5.

EXAMPLE

The sequence of terms together with their corresponding systems begins:

1: {}

2: {{}}

26: {{},{1,2}}

754: {{},{1,2},{1,3}}

32422: {{},{1,2},{1,3},{1,4}}

1523834: {{},{1,2},{1,3},{1,4},{2,3}}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

dae=Select[Range[100000], SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#], _?PrimeQ]&];

Table[dae[[Position[PrimeOmega/@dae, k][[1, 1]]]], {k, First[Split[Union[PrimeOmega/@dae], #2==#1+1&]]}]

CROSSREFS

MM-numbers of sets of sets with no singletons are A329630.

The case without empty edges is A329554.

MM-numbers of sets of sets are A302494.

Cf. A056239, A112798, A302242, A329552, A329557, A329558.

Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

Sequence in context: A156211 A156212 A138524 * A316747 A354244 A059516

Adjacent sequences: A329553 A329554 A329555 * A329557 A329558 A329559

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 17 2019

STATUS

approved