A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

1, 35, 141, 1713, 28011, 355, 34567, 4045, 54849, 64615, 15265, 95363, 126841

Offset

1,2

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=1..13.

Example

The sequence of terms together with their corresponding multisets of multisets begins:
       1: {}
      35: {{2},{1,1}}
     141: {{1},{2,3}}
     355: {{2},{1,1,3}}
    1713: {{1},{2,3,4}}
    4045: {{2},{1,1,3,4}}
   15265: {{2},{1,4},{1,1,3}}
   28011: {{1},{2,3,4,5}}
   34567: {{1,2},{3,4,5}}
   54849: {{1},{2,3},{4,5}}
   64615: {{2},{1,1,3,4,5}}
   95363: {{2,3},{1,1,4,5}}
  126841: {{3},{1,2},{1,4,5}}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule, Table[{p[[i]], i}, {i, Length[p]}], {1}])], {p, Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]], {n, 1000}];
Table[Position[dv, i][[1, 1]], {i, First/@Gather[dv]}]

Crossrefs

Sorted positions of first appearances in A330098.

The unsorted version is A330230.

The BII-number version is A330218.

MM-numbers of achiral multisets of multisets are A330232.

MM-numbers of fully-chiral multisets of multisets are A330236.

Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231.

Sequence in context: A350205 A267565 A330230 * A220014 A157286 A354543

Adjacent sequences: A330230 A330231 A330232 * A330234 A330235 A330236

Keyword

nonn,more

Author

Gus Wiseman, Dec 09 2019

Status

approved