A340018 MM-numbers of labeled graphs with half-loops covering an initial interval of positive integers, without isolated vertices.

1, 3, 13, 15, 39, 65, 141, 143, 145, 165, 195, 377, 429, 435, 611, 705, 715, 1131, 1363, 1551, 1595, 1833, 1885, 1937, 2021, 2117, 2145, 2235, 2365, 2397, 2409, 2431, 2465, 2805, 3055, 4089, 4147, 4785, 5655, 5811, 6063, 6149, 6235, 6351, 6409, 6721, 6815

Offset

1,2

Comments

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.

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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Also products of distinct primes whose prime indices are either themselves prime or a squarefree semiprime, and whose prime indices together cover an initial interval of positive integers. A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Links

Table of n, a(n) for n=1..47.

Example

The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}
     3: {{1}}
    13: {{1,2}}
    15: {{1},{2}}
    39: {{1},{1,2}}
    65: {{2},{1,2}}
   141: {{1},{2,3}}
   143: {{3},{1,2}}
   145: {{2},{1,3}}
   165: {{1},{2},{3}}
   195: {{1},{2},{1,2}}
   377: {{1,2},{1,3}}
   429: {{1},{3},{1,2}}
   435: {{1},{2},{1,3}}
   611: {{1,2},{2,3}}
   705: {{1},{2},{2,3}}
   715: {{2},{3},{1,2}}
  1131: {{1},{1,2},{1,3}}

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[1000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

Crossrefs

The version with full loops is A320461.

The version not necessarily covering an initial interval is A340019.

MM-numbers of graphs with loops are A340020.

A006450 lists primes of prime index.

A106349 lists primes of semiprime index.

A257994 counts prime prime indices.

A302242 is the weight of the multiset of multisets with MM-number n.

A302494 lists MM-numbers of sets of sets, with connected case A328514.

A309356 lists MM-numbers of simple graphs.

A322551 lists primes of squarefree semiprime index.

A339112 lists MM-numbers of multigraphs with loops.

A339113 lists MM-numbers of multigraphs.

Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

Sequence in context: A353060 A152270 A032919 * A216044 A023144 A152269

Adjacent sequences: A340015 A340016 A340017 * A340019 A340020 A340021

Keyword

nonn

Author

Gus Wiseman, Jan 02 2021

Status

approved