A330232 MM-numbers of achiral multisets of multisets.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80

Offset

1,2

Comments

First differs from A322554 in lacking 141.

A multiset of multisets is achiral if it is not changed by any permutation of the vertices.

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..67.

Example

The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}
  45: {{1},{1},{2}}
  61: {{1,2,2}}
  65: {{2},{1,2}}
  69: {{1},{2,2}}
  70: {{},{2},{1,1}}
  71: {{1,1,3}}
  74: {{},{1,1,2}}
  75: {{1},{2},{2}}
  77: {{1,1},{3}}
  78: {{},{1},{1,2}}
  87: {{1},{1,3}}
  89: {{1,1,1,2}}
  90: {{},{1},{1},{2}}

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]}]]
Select[Range[100], Length[graprms[primeMS/@primeMS[#]]]==1&]

Crossrefs

The fully-chiral version is A330236.

Achiral set-systems are counted by A083323.

MG-numbers of planted achiral trees are A214577.

MM-weight is A302242.

MM-numbers of costrict (or T_0) multisets of multisets are A322847.

BII-numbers of achiral set-systems are A330217.

Non-isomorphic achiral multiset partitions are A330223.

Achiral integer partitions are counted by A330224.

Achiral factorizations are A330234.

Cf. A001055, A003238, A007716, A056239, A112798, A303975, A330098, A330230, A330233.

Sequence in context: A132145 A272284 A322554 * A273882 A277112 A291168

Adjacent sequences: A330229 A330230 A330231 * A330233 A330234 A330235

Keyword

nonn

Author

Gus Wiseman, Dec 08 2019

Status

approved