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

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

Table of n, a(n) for n=1..46.

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[#])]&]

Crossrefs

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A305052, A305078.

Cf. A320456, A320458, A320459, A320461, A320533.

Sequence in context: A153119 A166703 A116522 * A108056 A018562 A112540

Adjacent sequences: A320459 A320460 A320461 * A320463 A320464 A320465

Keyword

nonn

Author

Gus Wiseman, Oct 13 2018

Status

approved