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A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

%I A320461 #11 Feb 16 2025 08:33:56
%S A320461 1,7,13,91,161,299,329,377,611,667,1261,1363,1937,2021,2093,2117,2639,
%T A320461 4277,4669,7567,8671,8827,9541,13559,14053,14147,14819,15617,16211,
%U A320461 17719,23989,24017,26273,27521,28681,29003,31349,31913,36569,44551,44603,46483,48691
%N A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.
%C A320461 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 A320461 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimpleGraph.html">Simple Graph</a>
%e A320461 The sequence of terms together with their multiset multisystems begins:
%e A320461      1: {}
%e A320461      7: {{1,1}}
%e A320461     13: {{1,2}}
%e A320461     91: {{1,1},{1,2}}
%e A320461    161: {{1,1},{2,2}}
%e A320461    299: {{2,2},{1,2}}
%e A320461    329: {{1,1},{2,3}}
%e A320461    377: {{1,2},{1,3}}
%e A320461    611: {{1,2},{2,3}}
%e A320461    667: {{2,2},{1,3}}
%e A320461   1261: {{3,3},{1,2}}
%e A320461   1363: {{1,3},{2,3}}
%e A320461   1937: {{1,2},{3,4}}
%e A320461   2021: {{1,4},{2,3}}
%e A320461   2093: {{1,1},{2,2},{1,2}}
%e A320461   2117: {{1,3},{2,4}}
%e A320461   2639: {{1,1},{1,2},{1,3}}
%e A320461   4277: {{1,1},{1,2},{2,3}}
%e A320461   4669: {{1,1},{2,2},{1,3}}
%t A320461 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320461 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320461 Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]
%Y A320461 Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.
%Y A320461 Cf. A320456, A320458, A320459, A320462, A320532.
%K A320461 nonn
%O A320461 1,2
%A A320461 _Gus Wiseman_, Oct 13 2018