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.

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A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 35, 49, 51, 53, 57, 59, 63, 64, 67, 73, 77, 81, 83, 85, 95, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 149, 153, 159, 161, 171, 175, 177, 187, 189, 201, 209, 217, 227, 233, 241, 243, 245
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

Examples

			The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  35: (((o))(oo))
  49: ((oo)(oo))
  51: ((o)((oo)))
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  63: ((o)(o)(oo))

Crossrefs

Cf. A000081, A007097, A290689, A303431, A304360, A306844, A316502, A318185, A318186.
Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324769.
Cf. A324838, A324840, A324844.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={},And@@fratQ/@primeMS[n]];
Select[Range[100],fratQ]

A324845 Matula-Goebel numbers of rooted trees where the branches of no non-leaf branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 38, 40, 43, 44, 46, 49, 50, 51, 53, 57, 58, 59, 62, 63, 64, 67, 68, 69, 70, 71, 73, 76, 77, 79, 80, 81, 83, 85, 86, 87, 88, 92, 93, 95, 97, 98, 99, 100, 103, 106
Offset: 1

Views

Author

Gus Wiseman, Mar 18 2019

Keywords

Examples

			The sequence of terms together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  14: (o(oo))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))

Links

Crossrefs

Cf. A000081, A007097, A290822, A306844, A318186.
Cf. A324694, A324738, A324744, A324749, A324754, A324759, A324765, A324768, A324838, A324842, A324844, A324846, A324847, A324849.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[!Divisible[n,x],{x,DeleteCases[primeMS[n],1]}],And@@qaQ/@primeMS[n]];
Select[Range[100],qaQ]

A317964 Prime numbers in the lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence (A304360).

Original entry on oeis.org

2, 5, 13, 17, 23, 31, 37, 43, 47, 61, 67, 73, 79, 89, 103, 107, 109, 113, 137, 149, 151, 163, 167, 179, 181, 193, 197, 223, 227, 233, 241, 251, 257, 263, 269, 271, 277, 281, 307, 317, 347, 349, 353, 359, 379, 383, 389, 397, 419, 421, 431, 433, 449, 457, 463, 467, 487, 499, 503, 509, 521, 523, 547
Offset: 1

Views

Author

N. J. A. Sloane, Aug 26 2018

Keywords

Comments

Also primes whose prime index is not in A304360, or is in A324696. A prime index of n is a number m such that prime(m) divides n. - Gus Wiseman, Mar 19 2019

Links

Crossrefs

Cf. A000720, A001462, A007097, A060197, A079254, A112798, A276625, A277098, A290822, A304360, A306844.
Cf. A324695, A324698, A324704, A324741, A324756, A324758, A324765, A324766, A324768.

Programs

  • Maple
    count:= 0:
P:= {}: A:= NULL:
for n from 2 while count < 100 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    P:= P union {ithprime(n)};
    if isprime(n) then A:= A, n; count:= count+1 fi;
  fi
od:
A; # Robert Israel, Aug 26 2018
  • Mathematica
    aQ[n_]:=n==1||Or@@Cases[FactorInteger[n],{p_,_}:>!aQ[PrimePi[p]]];
Prime[Select[Range[100],aQ]] (* Gus Wiseman, Mar 19 2019 *)

A358460 Number of locally disjoint ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 103, 301, 902, 2767, 8637, 27324, 87409, 282319, 919352
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2022

Keywords

Comments

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex.

Examples

			The a(1) = 1 through a(5) = 13 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (o(o))   ((ooo))
                 (((o)))  (o(o)o)
                          (o(oo))
                          (oo(o))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o(o)))
                          (o((o)))
                          ((((o))))

Crossrefs

The locally non-intersecting version is A143363, unordered A007562.
The unordered version is A316473, ranked by A316495.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A006013, A302696, A316471, A316694, A318185, A319378, A324768, A324844.

Programs

  • Mathematica
    aot[n_]:=If[n==1,{{}},Join @@ Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,{_,x_,_},_,{_,x_,_},_}]&]],{n,10}]
Previous Showing 11-14 of 14 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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