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.

Showing 1-4 of 4 results.

A316795 Number of aperiodic rooted trees on n nodes with locally distinct multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 8, 17, 30, 55, 101, 194, 352, 663, 1227, 2275, 4225, 7877, 14600, 27158, 50414, 93666, 173972, 323286, 600353, 1115407, 2071843, 3848794, 7149196, 13280874, 24669606, 45827047, 85126845, 158131764, 293742200, 545655290, 1013598733
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

An aperiodic rooted tree is an unlabeled rooted tree in which the multiplicities of branches under any given node are relatively prime. A rooted tree has locally distinct multiplicities if the multiset of branches under any given node has all distinct multiplicities.

Examples

			The a(7) = 8 trees:
((((((o))))))
(((oo(o))))
((oo((o))))
((o(o)(o)))
((ooo(o)))
(oo(((o))))
(ooo((o)))
(oooo(o))

Links

Crossrefs

Cf. A000081, A000837, A004111, A301700, A303431, A316793, A316794, A316796.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],And[UnsameQ@@Length/@Split[#],GCD@@Length/@Split[#]==1]&]];
Table[Length[strut[n]],{n,15}]
  • PARI
    C(v,n)={my(recurse(r,b,g,p,k)=if(!r, g==1, sum(m=1, r, if(!bittest(b,m), sum(i=1, min(r\m, p), my(f=if(i==p, k+1, 1)); if(v[i]>=f, (v[i]-f+1)*self()(r-m*i, bitor(b, 1<Andrew Howroyd, Feb 08 2020

Extensions

Terms a(26) and beyond from Andrew Howroyd, Feb 08 2020

A316793 Numbers whose prime multiplicities are distinct and relatively prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 124, 127, 131
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

A subsequence of A007916.

Examples

			60 = 2^2 * 3^1 * 5^1 has prime multiplicities (2,1,1), which are relatively prime but not distinct, so 60 does not belong to the sequence.
72 = 2^3 * 3^2 has prime multiplicities (3,2), which are distinct and relatively prime, so 72 belongs to the sequence.
144 = 2^4 * 3^2 has prime multiplicities (4,2), which are distinct but not relatively prime, so 144 does not belong to the sequence.

Crossrefs

Cf. A000837, A005117, A007916, A130091, A301700, A303431, A316794, A316795, A316796.

Programs

  • Mathematica
    Select[Range[100],And[UnsameQ@@Last/@FactorInteger[#],GCD@@Last/@FactorInteger[#]==1]&]

A316794 Matula-Goebel numbers of aperiodic rooted trees with locally distinct multiplicities.

Original entry on oeis.org

1, 2, 3, 5, 11, 12, 18, 20, 24, 31, 37, 40, 44, 45, 48, 50, 54, 61, 71, 72, 75, 80, 88, 89, 96, 99, 108, 124, 127, 135, 148, 157, 160, 162, 173, 176, 192, 193, 197, 200, 223, 229, 242, 244, 248, 250, 251, 275, 279, 283, 284, 288, 296, 297, 320, 333, 352, 353
Offset: 1

Views

Author

Gus Wiseman, Jul 14 2018

Keywords

Comments

A positive integer belongs to the sequence iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) as well as to A130091 (numbers whose prime multiplicities are distinct), and all of its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.

Examples

			Sequence of aperiodic rooted trees with locally distinct multiplicities preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
  11: ((((o))))
  12: (oo(o))
  18: (o(o)(o))
  20: (oo((o)))
  24: (ooo(o))
  31: (((((o)))))
  37: ((oo(o)))
  40: (ooo((o)))
  44: (oo(((o))))
  45: ((o)(o)((o)))
  48: (oooo(o))
  50: (o((o))((o)))

Crossrefs

Cf. A000081, A004111, A007097, A007916, A061775, A276625, A301700 A303431, A316793, A316795, A316796.

Programs

  • Mathematica
    mgsbQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],GCD@@Last/@FactorInteger[n]==1,And@@Cases[FactorInteger[n],{p_,_}:>mgsbQ[PrimePi[p]]]]];
Select[Range[100],mgsbQ]

A319272 Numbers whose prime multiplicities are distinct and whose prime indices are term of the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 31, 32, 37, 40, 44, 45, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 75, 76, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 103, 107, 108, 112, 121, 124, 125, 127, 128, 131, 135, 136, 144, 147, 148
Offset: 1

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n.
Also Matula-Goebel numbers of rooted trees in which the multiplicities in the multiset of branches directly under any given node are distinct.

Examples

			36 is not in the sequence because 36 = 2^2 * 3^2 does not have distinct prime multiplicities.
The sequence of terms of the sequence followed by their Matula-Goebel trees begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  12: (oo(o))
  16: (oooo)
  17: (((oo)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((o)))
  23: (((o)(o)))
  24: (ooo(o))
  25: (((o))((o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  31: (((((o)))))

Links

Crossrefs

Cf. A000081, A004111, A007097, A061775, A098859, A130091, A255231, A276625, A316793, A316794, A316795, A316796.

Programs

  • Mathematica
    mgsiQ[n_]:=Or[n==1,And[UnsameQ@@Last/@FactorInteger[n],And@@Cases[FactorInteger[n],{p_,_}:>mgsiQ[PrimePi[p]]]]];
Select[Range[100],mgsiQ]
  • PARI
    is(n)={my(f=factor(n)); if(#Set(f[,2])<#f~, 0, for(i=1, #f~, if(!is(primepi(f[i,1])), return(0))); 1)}
{ select(is, [1..200]) } \\ Andrew Howroyd, Mar 01 2020

Extensions

Terms a(53) and beyond from Andrew Howroyd, Mar 01 2020
Showing 1-4 of 4 results.

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