A330346
Number of unlabeled simple graphs covering n vertices with exactly two automorphisms.
Original entry on oeis.org
0, 0, 1, 1, 2, 9, 37
Offset: 0
Non-isomorphic representatives of the a(5) = 9 graphs:
{12,13,14,25}
{12,13,24,35}
{12,13,14,23,25}
{12,13,14,23,45}
{12,13,15,24,34}
{12,13,14,15,23,24}
{12,13,14,23,24,35}
{12,13,14,23,25,45}
{12,13,14,15,23,24,35}
The non-covering version is
A330344.
Unlabeled covering graphs are
A002494.
Unlabeled connected graphs with exactly two automorphisms are
A241454.
Cf.
A000088,
A003400,
A055621,
A124059,
A330098,
A330227,
A330230,
A330231,
A330294,
A330295,
A330343.
A330343
Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.
Original entry on oeis.org
1, 0, 0, 0, 0, 5760, 766080, 149022720, 48990251520, 28928242022400, 32147584690636800, 69035206021583155200
Offset: 1
Covering simple graphs are
A006129.
Full chiral integer partitions are
A330228.
Fully chiral factorizations are
A330235.
Cf.
A006125,
A016031,
A124059,
A143543,
A330098,
A330224,
A330226,
A330227,
A330230,
A330231,
A330236.
-
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==n!&]],{n,5}] (* brute force, not for computation *)
A330281
Numbers whose prime-indices do not have weakly increasing numbers of distinct prime factors.
Original entry on oeis.org
221, 247, 299, 403, 442, 494, 533, 598, 663, 689, 741, 767, 806, 871, 884, 897, 899, 988, 1066, 1079, 1105, 1189, 1196, 1209, 1235, 1261, 1326, 1339, 1378, 1417, 1482, 1495, 1517, 1534, 1537, 1547, 1599, 1612, 1651, 1703, 1711, 1729, 1742, 1768, 1794, 1798
Offset: 1
The sequence of terms together with their prime indices begins:
221: {6,7}
247: {6,8}
299: {6,9}
403: {6,11}
442: {1,6,7}
494: {1,6,8}
533: {6,13}
598: {1,6,9}
663: {2,6,7}
689: {6,16}
741: {2,6,8}
767: {6,17}
806: {1,6,11}
871: {6,19}
884: {1,1,6,7}
For example, 884 has prime indices {1,1,6,7} with numbers of distinct prime factors (0,0,2,1), which is not weakly increasing, so 884 belongs to the sequence.
The version where prime factors are counted with multiplicity is
A330103.
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