cp's OEIS Frontend

A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

%I A320459 #10 Feb 16 2025 08:33:56
%S A320459 1,13,169,377,611,1363,1937,2021,2117,2197,4901,7943,10933,16211,
%T A320459 17719,25181,26273,27521,28561,28717,39527,44603,56173,58609,61393,
%U A320459 63713,64061,83291,86903,91031,91039,94987,99499,103259,141401,142129,143663,146653,147533
%N A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.
%C A320459 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 A320459 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%e A320459 The sequence of terms together with their multiset multisystems begins:
%e A320459       1: {}
%e A320459      13: {{1,2}}
%e A320459     169: {{1,2},{1,2}}
%e A320459     377: {{1,2},{1,3}}
%e A320459     611: {{1,2},{2,3}}
%e A320459    1363: {{1,3},{2,3}}
%e A320459    1937: {{1,2},{3,4}}
%e A320459    2021: {{1,4},{2,3}}
%e A320459    2117: {{1,3},{2,4}}
%e A320459    2197: {{1,2},{1,2},{1,2}}
%e A320459    4901: {{1,2},{1,2},{1,3}}
%e A320459    7943: {{1,2},{1,2},{2,3}}
%e A320459   10933: {{1,2},{1,3},{1,3}}
%e A320459   16211: {{1,2},{1,3},{1,4}}
%e A320459   17719: {{1,2},{1,3},{2,3}}
%e A320459   25181: {{1,2},{1,2},{3,4}}
%e A320459   26273: {{1,2},{1,4},{2,3}}
%e A320459   27521: {{1,2},{1,3},{2,4}}
%e A320459   28561: {{1,2},{1,2},{1,2},{1,2}}
%e A320459   28717: {{1,2},{2,3},{2,3}}
%e A320459   39527: {{1,3},{1,3},{2,3}}
%e A320459   44603: {{1,2},{2,3},{2,4}}
%t A320459 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320459 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320459 Select[Range[100000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]
%Y A320459 Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052, A305078.
%Y A320459 Cf. A320456, A320458, A320461, A320462, A320464.
%K A320459 nonn
%O A320459 1,2
%A A320459 _Gus Wiseman_, Oct 13 2018