

A330232


MMnumbers of achiral multisets of multisets.


14



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
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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 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=1..67.


EXAMPLE

The sequence of nonachiral multisets of multisets (the complement of this sequence) together with their MMnumbers 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 fullychiral version is A330236.
Achiral setsystems are counted by A083323.
MGnumbers of planted achiral trees are A214577.
MMweight is A302242.
MMnumbers of costrict (or T_0) multisets of multisets are A322847.
BIInumbers of achiral setsystems are A330217.
Nonisomorphic 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



