A340018 - OEIS

%I #10 Jan 03 2021 15:53:15

%S 1,3,13,15,39,65,141,143,145,165,195,377,429,435,611,705,715,1131,

%T 1363,1551,1595,1833,1885,1937,2021,2117,2145,2235,2365,2397,2409,

%U 2431,2465,2805,3055,4089,4147,4785,5655,5811,6063,6149,6235,6351,6409,6721,6815

%N MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

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

%C 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 Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

%e The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:

%e      1: {}

%e      3: {{1}}

%e     13: {{1,2}}

%e     15: {{1},{2}}

%e     39: {{1},{1,2}}

%e     65: {{2},{1,2}}

%e    141: {{1},{2,3}}

%e    143: {{3},{1,2}}

%e    145: {{2},{1,3}}

%e    165: {{1},{2},{3}}

%e    195: {{1},{2},{1,2}}

%e    377: {{1,2},{1,3}}

%e    429: {{1},{3},{1,2}}

%e    435: {{1},{2},{1,3}}

%e    611: {{1,2},{2,3}}

%e    705: {{1},{2},{2,3}}

%e    715: {{2},{3},{1,2}}

%e   1131: {{1},{1,2},{1,3}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];

%t Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

%Y The version with full loops is A320461.

%Y The version not necessarily covering an initial interval is A340019.

%Y MM-numbers of graphs with loops are A340020.

%Y A006450 lists primes of prime index.

%Y A106349 lists primes of semiprime index.

%Y A257994 counts prime prime indices.

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

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

%Y A309356 lists MM-numbers of simple graphs.

%Y A322551 lists primes of squarefree semiprime index.

%Y A339112 lists MM-numbers of multigraphs with loops.

%Y A339113 lists MM-numbers of multigraphs.

%Y Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 02 2021