cp's OEIS Frontend

A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.

%I A320458 #11 Feb 16 2025 08:33:56
%S A320458 1,13,377,611,1363,1937,2021,2117,16211,17719,26273,27521,44603,56173,
%T A320458 58609,83291,91031,91039,99499,141401,143663,146653,147533,153023,
%U A320458 159659,167243,170839,203087,237679,243893,265369,271049,276877,290029,301129,315433,467711
%N A320458 MM-numbers of labeled simple graphs spanning an initial interval of positive integers.
%C A320458 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 multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}.
%H A320458 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%e A320458 The sequence of terms together with their multiset multisystems begins:
%e A320458       1: {}
%e A320458      13: {{1,2}}
%e A320458     377: {{1,2},{1,3}}
%e A320458     611: {{1,2},{2,3}}
%e A320458    1363: {{1,3},{2,3}}
%e A320458    1937: {{1,2},{3,4}}
%e A320458    2021: {{1,4},{2,3}}
%e A320458    2117: {{1,3},{2,4}}
%e A320458   16211: {{1,2},{1,3},{1,4}}
%e A320458   17719: {{1,2},{1,3},{2,3}}
%e A320458   26273: {{1,2},{1,4},{2,3}}
%e A320458   27521: {{1,2},{1,3},{2,4}}
%e A320458   44603: {{1,2},{2,3},{2,4}}
%e A320458   56173: {{1,2},{1,3},{3,4}}
%e A320458   58609: {{1,3},{1,4},{2,3}}
%e A320458   83291: {{1,2},{1,4},{3,4}}
%e A320458   91031: {{1,3},{1,4},{2,4}}
%e A320458   91039: {{1,2},{2,3},{3,4}}
%e A320458   99499: {{1,3},{2,3},{2,4}}
%t A320458 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320458 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320458 Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]
%Y A320458 Cf. A001222, A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A302491, A305052.
%Y A320458 Cf. A320456, A320459, A320461, A320462, A320463.
%K A320458 nonn
%O A320458 1,2
%A A320458 _Gus Wiseman_, Oct 13 2018