%I #10 Dec 29 2018 09:18:42

%S 1,7,13,91,161,299,329,377,611,667,1261,1363,1937,2021,2093,2117,2639,

%T 4277,4669,7567,8671,8827,9541,13559,14053,14147,14819,15617,16211,

%U 17719,23989,24017,26273,27521,28681,29003,31349,31913,36569,44551,44603,46483,48691

%N MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

%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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>

%e The sequence of terms together with their multiset multisystems begins:

%e 1: {}

%e 7: {{1,1}}

%e 13: {{1,2}}

%e 91: {{1,1},{1,2}}

%e 161: {{1,1},{2,2}}

%e 299: {{2,2},{1,2}}

%e 329: {{1,1},{2,3}}

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

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

%e 667: {{2,2},{1,3}}

%e 1261: {{3,3},{1,2}}

%e 1363: {{1,3},{2,3}}

%e 1937: {{1,2},{3,4}}

%e 2021: {{1,4},{2,3}}

%e 2093: {{1,1},{2,2},{1,2}}

%e 2117: {{1,3},{2,4}}

%e 2639: {{1,1},{1,2},{1,3}}

%e 4277: {{1,1},{1,2},{2,3}}

%e 4669: {{1,1},{2,2},{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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

%Y Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

%Y Cf. A320456, A320458, A320459, A320462, A320532.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 13 2018