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A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

%I A320462 #13 Feb 16 2025 08:33:56
%S A320462 1,7,13,49,91,161,169,299,329,343,377,611,637,667,1127,1183,1261,1363,
%T A320462 1937,2021,2093,2117,2197,2303,2401,2639,3703,3887,4277,4459,4669,
%U A320462 4901,6877,7567,7889,7943,8281,8671,8827,9541,10933,13559,14053,14147,14651,14819
%N A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.
%C A320462 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 A320462 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%e A320462 The sequence of terms together with their multiset multisystems begins:
%e A320462      1: {}
%e A320462      7: {{1,1}}
%e A320462     13: {{1,2}}
%e A320462     49: {{1,1},{1,1}}
%e A320462     91: {{1,1},{1,2}}
%e A320462    161: {{1,1},{2,2}}
%e A320462    169: {{1,2},{1,2}}
%e A320462    299: {{2,2},{1,2}}
%e A320462    329: {{1,1},{2,3}}
%e A320462    343: {{1,1},{1,1},{1,1}}
%e A320462    377: {{1,2},{1,3}}
%e A320462    611: {{1,2},{2,3}}
%e A320462    637: {{1,1},{1,1},{1,2}}
%e A320462    667: {{2,2},{1,3}}
%e A320462   1127: {{1,1},{1,1},{2,2}}
%e A320462   1183: {{1,1},{1,2},{1,2}}
%t A320462 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320462 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320462 Select[Range[10000],And[normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]
%Y A320462 Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A305052, A305078.
%Y A320462 Cf. A320456, A320458, A320459, A320461, A320533.
%K A320462 nonn
%O A320462 1,2
%A A320462 _Gus Wiseman_, Oct 13 2018