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A320462
MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.
12
1, 7, 13, 49, 91, 161, 169, 299, 329, 343, 377, 611, 637, 667, 1127, 1183, 1261, 1363, 1937, 2021, 2093, 2117, 2197, 2303, 2401, 2639, 3703, 3887, 4277, 4459, 4669, 4901, 6877, 7567, 7889, 7943, 8281, 8671, 8827, 9541, 10933, 13559, 14053, 14147, 14651, 14819
OFFSET
1,2
COMMENTS
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}}.
LINKS
Eric Weisstein's World of Mathematics, Simple Graph
EXAMPLE
The sequence of terms together with their multiset multisystems begins:
1: {}
7: {{1,1}}
13: {{1,2}}
49: {{1,1},{1,1}}
91: {{1,1},{1,2}}
161: {{1,1},{2,2}}
169: {{1,2},{1,2}}
299: {{2,2},{1,2}}
329: {{1,1},{2,3}}
343: {{1,1},{1,1},{1,1}}
377: {{1,2},{1,3}}
611: {{1,2},{2,3}}
637: {{1,1},{1,1},{1,2}}
667: {{2,2},{1,3}}
1127: {{1,1},{1,1},{2,2}}
1183: {{1,1},{1,2},{1,2}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[sys_]:=Or[Length[sys]==0, Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000], And[normQ[primeMS/@primeMS[#]], And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 13 2018
STATUS
approved