cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-14 of 14 results.

A317719 Numbers that are not powerful tree numbers.

Original entry on oeis.org

6, 10, 12, 13, 14, 15, 18, 20, 21, 22, 24, 26, 28, 29, 30, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 89, 90, 91
Offset: 1

Views

Author

Gus Wiseman, Aug 05 2018

Keywords

Comments

A positive integer n is a powerful tree number iff either n = 1 or n is a prime number whose prime index is a powerful tree number, or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all powerful tree numbers. A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of numbers that are not powerful tree numbers together with their Matula-Goebel trees begins:
   6: (o(o))
  10: (o((o)))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Crossrefs

Complement of A318612.
Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.
Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718.

Programs

  • Mathematica
    powgoQ[n_]:=Or[n==1,If[PrimeQ[n],powgoQ[PrimePi[n]],And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!powgoQ[#]&]

A317785 Number of locally connected rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 42, 55, 67, 91, 109, 144, 177, 228, 281, 366, 448, 579, 720, 916, 1142
Offset: 1

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Comments

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

Examples

			The a(11) = 12 locally connected rooted trees:
  ((((((((((o))))))))))
  ((((((((o)(o))))))))
  (((((((o))((o)))))))
  ((((((o)))(((o))))))
  (((((o))))((((o)))))
  ((((((o)(o)(o))))))
  (((((o))((o)(o)))))
  ((((o))((o))((o))))
  ((((o)(o)(o)(o))))
  (((o))((o)(o)(o)))
  (((o)(o))((o)(o)))
  ((o)(o)(o)(o)(o))

Links

Crossrefs

Cf. A000081, A276625, A286518, A286520, A301700, A304714, A316473, A316475, A317077, A317078, A317708, A317787.

Programs

  • Mathematica
    multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],multijoin@@s[[c[[1]]]]]]]]];
rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Length[csm[#]]==1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

Original entry on oeis.org

1, 1, 1, 3, 8, 26, 76, 247, 783, 2565, 8447, 28256, 95168, 323720, 1108415, 3821144, 13246307, 46158480, 161574043, 567925140, 2003653016, 7092953340, 25186731980, 89690452750, 320221033370, 1146028762599, 4110596336036, 14774346783745, 53203889807764, 191934931634880
Offset: 1

Views

Author

Gus Wiseman, Sep 05 2018

Keywords

Comments

Also the number of plane trees with n nodes where the sequence of branches directly under any given node has relatively prime run-lengths.

Examples

			The a(5) = 8 locally aperiodic plane trees:
  ((((o)))),
  (((o)o)), ((o(o))), (((o))o), (o((o))),
  ((o)oo), (o(o)o), (oo(o)).
The a(6) = 26 locally aperiodic plane trees:
  (((((o)))))  ((((o)o)))  (((o)oo))  ((o)ooo)
               (((o(o))))  ((o(o)o))  (o(o)oo)
               ((((o))o))  ((oo(o)))  (oo(o)o)
               ((o((o))))  (((o)o)o)  (ooo(o))
               ((((o)))o)  ((o(o))o)
               (o(((o))))  (o((o)o))
               (((o))(o))  (o(o(o)))
               ((o)((o)))  (((o))oo)
                           (o((o))o)
                           (oo((o)))
                           ((o)(o)o)
                           ((o)o(o))
                           (o(o)(o))

Links

Crossrefs

Cf. A000108, A000837, A007853, A032171, A032200, A254040, A301700, A303386, A303431, A304173, A304175, A317708, A317852.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[aperplane[n]],{n,10}]
  • PARI
    Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ Andrew Howroyd, Feb 08 2020

Extensions

a(16)-a(17) from Robert Price, Sep 15 2018
Terms a(18) and beyond from Andrew Howroyd, Feb 08 2020

A317720 Numbers that are not uniform relatively prime tree numbers.

Original entry on oeis.org

9, 12, 18, 20, 21, 23, 24, 25, 27, 28, 37, 39, 40, 44, 45, 46, 48, 49, 50, 52, 54, 56, 57, 60, 61, 63, 65, 68, 69, 71, 72, 73, 74, 75, 76, 80, 81, 83, 84, 87, 88, 89, 90, 91, 92, 96, 97, 98, 99, 103, 104, 107, 108, 111, 112, 115, 116, 117, 120, 121, 122, 124
Offset: 1

Views

Author

Gus Wiseman, Aug 05 2018

Keywords

Comments

A positive integer n is a uniform relatively prime tree number iff either n = 1 or n is a prime number whose prime index is a uniform relatively prime tree number, or n is a power of a squarefree number whose prime indices are relatively prime and are themselves uniform relatively prime tree numbers. A prime index of n is a number m such that prime(m) divides n.

Examples

			The sequence of non-uniform tree numbers together with their Matula-Goebel trees begins:
   9: ((o)(o))
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  21: ((o)(oo))
  23: (((o)(o)))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  37: ((oo(o)))
  39: ((o)(o(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))

Crossrefs

Cf. A061775, A072774, A111299, A214577, A276625, A277098, A303431, A317590.
Cf. A317705, A317707, A317708, A317709, A317710, A317712, A317717, A317718.

Programs

  • Mathematica
    rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[SameQ@@FactorInteger[n][[All,2]],GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[200],!rupQ[#]&]
Previous Showing 11-14 of 14 results.

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