cp's OEIS Frontend

A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

%I A340020 #5 Jan 03 2021 15:53:28
%S A340020 1,7,13,23,29,43,47,73,79,91,97,101,137,139,149,161,163,167,199,203,
%T A340020 227,233,257,269,271,293,299,301,313,329,347,373,377,389,421,439,443,
%U A340020 449,467,487,491,499,511,553,559,577,607,611,631,647,653,661,667,673,677
%N A340020 MM-numbers of labeled graphs with loops, without isolated vertices.
%C A340020 Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.
%C A340020 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 A340020 Also products of distinct primes whose prime indices are semiprimes, where a semiprime (A001358) is a product of any two prime numbers.
%e A340020 The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
%e A340020       1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
%e A340020       7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
%e A340020      13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
%e A340020      23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
%e A340020      29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
%e A340020      43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
%e A340020      47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
%e A340020      73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
%e A340020      79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
%e A340020      91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
%e A340020      97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
%e A340020     101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
%e A340020     137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
%e A340020     139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
%e A340020     149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}
%t A340020 Select[Range[100],SquareFreeQ[#]&&FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;PrimeOmega[PrimePi[p]]!=2]&]
%Y A340020 The case with only one edge is A106349.
%Y A340020 The case covering an initial interval is A320461.
%Y A340020 The version allowing multiple edges is A339112.
%Y A340020 The half-loop version covering an initial interval is A340018.
%Y A340020 The half-loop version is A340019.
%Y A340020 A006450 lists primes of prime index.
%Y A340020 A302242 is the weight of the multiset of multisets with MM-number n.
%Y A340020 A302494 lists MM-numbers of sets of sets, with connected case A328514.
%Y A340020 A309356 lists MM-numbers of simple graphs.
%Y A340020 A339113 lists MM-numbers of multigraphs.
%Y A340020 Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.
%K A340020 nonn
%O A340020 1,2
%A A340020 _Gus Wiseman_, Jan 02 2021