A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 39, 41, 43, 47, 51, 55, 59, 65, 67, 73, 79, 83, 85, 87, 93, 101, 109, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237, 241, 249

Offset: 1

Author

Gus Wiseman, Jan 02 2021

Keywords

• nonn

Comments

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal 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}}.

Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime (A006881).

Examples

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}              55: {{2},{3}}      137: {{2,5}}
     3: {{1}}           59: {{7}}          139: {{1,7}}
     5: {{2}}           65: {{2},{1,2}}    141: {{1},{2,3}}
    11: {{3}}           67: {{8}}          143: {{3},{1,2}}
    13: {{1,2}}         73: {{2,4}}        145: {{2},{1,3}}
    15: {{1},{2}}       79: {{1,5}}        149: {{3,4}}
    17: {{4}}           83: {{9}}          155: {{2},{5}}
    29: {{1,3}}         85: {{2},{4}}      157: {{12}}
    31: {{5}}           87: {{1},{1,3}}    163: {{1,8}}
    33: {{1},{3}}       93: {{1},{5}}      165: {{1},{2},{3}}
    39: {{1},{1,2}}    101: {{1,6}}        167: {{2,6}}
    41: {{6}}          109: {{10}}         177: {{1},{7}}
    43: {{1,4}}        123: {{1},{6}}      179: {{13}}
    47: {{2,3}}        127: {{11}}         187: {{3},{4}}
    51: {{1},{4}}      129: {{1},{1,4}}    191: {{14}}
		

Crossrefs

The version with full loops covering an initial interval is A320461.

The case covering an initial interval is A340018.

The version with full loops is A340020.

A006450 lists primes of prime index.

A106349 lists primes of semiprime index.

A257994 counts prime prime indices.

A302242 is the weight of the multiset of multisets with MM-number n.

A302494 lists MM-numbers of sets of sets, with connected case A328514.

A309356 lists MM-numbers of simple graphs.

A322551 lists primes of squarefree semiprime index.

A330944 counts nonprime prime indices.

A339112 lists MM-numbers of multigraphs with loops.

A339113 lists MM-numbers of multigraphs.

Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]