cp's OEIS Frontend

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

%I A340019 #9 Jan 03 2021 15:53:23
%S A340019 1,3,5,11,13,15,17,29,31,33,39,41,43,47,51,55,59,65,67,73,79,83,85,87,
%T A340019 93,101,109,123,127,129,137,139,141,143,145,149,155,157,163,165,167,
%U A340019 177,179,187,191,195,199,201,205,211,215,219,221,233,235,237,241,249
%N A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.
%C A340019 Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.
%C A340019 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}}.
%C A340019 Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime (A006881).
%e A340019 The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
%e A340019      1: {}              55: {{2},{3}}      137: {{2,5}}
%e A340019      3: {{1}}           59: {{7}}          139: {{1,7}}
%e A340019      5: {{2}}           65: {{2},{1,2}}    141: {{1},{2,3}}
%e A340019     11: {{3}}           67: {{8}}          143: {{3},{1,2}}
%e A340019     13: {{1,2}}         73: {{2,4}}        145: {{2},{1,3}}
%e A340019     15: {{1},{2}}       79: {{1,5}}        149: {{3,4}}
%e A340019     17: {{4}}           83: {{9}}          155: {{2},{5}}
%e A340019     29: {{1,3}}         85: {{2},{4}}      157: {{12}}
%e A340019     31: {{5}}           87: {{1},{1,3}}    163: {{1,8}}
%e A340019     33: {{1},{3}}       93: {{1},{5}}      165: {{1},{2},{3}}
%e A340019     39: {{1},{1,2}}    101: {{1,6}}        167: {{2,6}}
%e A340019     41: {{6}}          109: {{10}}         177: {{1},{7}}
%e A340019     43: {{1,4}}        123: {{1},{6}}      179: {{13}}
%e A340019     47: {{2,3}}        127: {{11}}         187: {{3},{4}}
%e A340019     51: {{1},{4}}      129: {{1},{1,4}}    191: {{14}}
%t A340019 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A340019 Select[Range[1000],And[SquareFreeQ[#],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]
%Y A340019 The version with full loops covering an initial interval is A320461.
%Y A340019 The case covering an initial interval is A340018.
%Y A340019 The version with full loops is A340020.
%Y A340019 A006450 lists primes of prime index.
%Y A340019 A106349 lists primes of semiprime index.
%Y A340019 A257994 counts prime prime indices.
%Y A340019 A302242 is the weight of the multiset of multisets with MM-number n.
%Y A340019 A302494 lists MM-numbers of sets of sets, with connected case A328514.
%Y A340019 A309356 lists MM-numbers of simple graphs.
%Y A340019 A322551 lists primes of squarefree semiprime index.
%Y A340019 A330944 counts nonprime prime indices.
%Y A340019 A339112 lists MM-numbers of multigraphs with loops.
%Y A340019 A339113 lists MM-numbers of multigraphs.
%Y A340019 Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.
%K A340019 nonn
%O A340019 1,2
%A A340019 _Gus Wiseman_, Jan 02 2021