A320458 -id:A320458 - cp's OEIS frontend

A320635 MM-numbers of simple labeled connected graphs spanning an initial interval of positive integers.

13, 377, 611, 1363, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 147533, 203087, 301129, 315433, 467711, 761917, 1183403, 1280669, 1293487, 1917929, 2075567, 2174159, 2220907, 2415439, 2640131

Offset: 1

Gus Wiseman, Oct 18 2018

nonn

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}}.

			The sequence of terms together with their multiset multisystems begins:
       13: {{1,2}}
      377: {{1,2},{1,3}}
      611: {{1,2},{2,3}}
     1363: {{1,3},{2,3}}
    16211: {{1,2},{1,3},{1,4}}
    17719: {{1,2},{1,3},{2,3}}
    26273: {{1,2},{1,4},{2,3}}
    27521: {{1,2},{1,3},{2,4}}
    44603: {{1,2},{2,3},{2,4}}
    56173: {{1,2},{1,3},{3,4}}
    58609: {{1,3},{1,4},{2,3}}
    83291: {{1,2},{1,4},{3,4}}
    91031: {{1,3},{1,4},{2,4}}
    91039: {{1,2},{2,3},{3,4}}
    99499: {{1,3},{2,3},{2,4}}
   141401: {{1,2},{2,4},{3,4}}
   147533: {{1,4},{2,3},{2,4}}
   203087: {{1,3},{2,3},{3,4}}
   301129: {{1,4},{2,3},{3,4}}
   315433: {{1,3},{2,4},{3,4}}
   467711: {{1,4},{2,4},{3,4}}
   761917: {{1,2},{1,3},{1,4},{2,3}}
  1183403: {{1,2},{1,3},{1,4},{2,4}}
  1280669: {{1,2},{1,3},{1,4},{1,5}}

Cf. A001222, A007717, A055932, A056239, A112798, A255906, A290103, A302242, A302491, A305078, A320456, A320458.

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]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#]),Length[zsm[primeMS[#]]]==1]&]

A322552 MM-numbers of triangles.

Original entry on oeis.org

17719, 40807, 140699, 185803, 219271, 421031, 511219, 570011, 588787, 897689, 916777, 1321433, 1581827, 1654823, 1769609, 1854983, 2028181, 2358773, 2456737, 2943343, 3641501, 3705221, 3890389, 3902981, 4186793, 4807489, 5176613, 5263759, 5693197, 6308857, 6515111, 6566717

Offset: 1

Gus Wiseman, Dec 15 2018

Sequence consists of terms of the form prime(p*q) * prime(p*r) * prime(q*r), with p, q, and r distinct primes. - Charlie Neder, Dec 23 2018

			The sequence of triangles whose MM-numbers belong to the sequence begins:
   17719: {{1,2},{1,3},{2,3}}
   40807: {{1,2},{1,4},{2,4}}
  140699: {{1,2},{1,5},{2,5}}
  185803: {{1,3},{1,4},{3,4}}
  219271: {{1,2},{1,6},{2,6}}
  421031: {{1,2},{1,7},{2,7}}
  511219: {{2,3},{2,4},{3,4}}
  570011: {{1,2},{1,8},{2,8}}
  588787: {{1,3},{1,5},{3,5}}
  897689: {{1,2},{1,9},{2,9}}
  916777: {{1,3},{1,6},{3,6}}

David A. Corneth, Table of n, a(n) for n = 1..12878 (first 900 terms from Charlie Neder)

Cf. A001222, A001358, A003963, A006881, A056239, A106349, A112798, A302242, A305078, A320458, A320635, A322551.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100000],And[SquareFreeQ[#],PrimeOmega[#]==3,And@@(SquareFreeQ[#]&&PrimeOmega[#]==2&/@primeMS[#]),SameQ[##,2]&@@Length/@Split[Sort[Join@@primeMS/@primeMS[#]]]]&]

a(12)-a(32) from Charlie Neder, Dec 27 2018

A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

Original entry on oeis.org

1, 1, 2, 5, 18, 69, 390, 2703, 59474, 1548349, 168926258, 12165065351, 7074423247562, 2294426405580191, 4218009215702391954, 3810376434461484994317, 35102248193591661086921250, 156873334244228518638713087133, 4144940994226400702145709978234154

Offset: 0

Gus Wiseman, Dec 15 2018

nonn

Such graphs may be said to have regular support.

			The a(4) = 18 edge sets:
  {}
  {{1,2}}
  {{1,3}}
  {{1,4}}
  {{2,3}}
  {{2,4}}
  {{3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A000569, A005176, A038041, A059441, A295193, A306017, A306019, A319169, A320458.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],SameQ@@Length/@Split[Sort[Join@@#]]&]],{n,6}]

a(n) = 1 + Sum_{k=1..n} binomial(n, k)*(A295193(k) - 1). - Andrew Howroyd, Dec 17 2018

a(8)-a(15) from Andrew Howroyd, Dec 17 2018

a(16)-a(18) from Andrew Howroyd, May 21 2020

cp's OEIS Frontend

A320635 MM-numbers of simple labeled connected graphs spanning an initial interval of positive integers.

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A322552 MM-numbers of triangles.

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A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

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