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 1-5 of 5 results.

A198328 The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the leaves, together with their incident edges.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 5, 2, 3, 2, 6, 1, 3, 4, 2, 3, 4, 5, 7, 2, 9, 3, 8, 2, 5, 6, 11, 1, 10, 3, 6, 4, 3, 2, 6, 3, 5, 4, 3, 5, 12, 7, 13, 2, 4, 9, 6, 3, 2, 8, 15, 2, 4, 5, 5, 6, 7, 11, 8, 1, 9, 10, 3, 3, 14, 6, 5, 4, 7, 3, 18, 2, 10, 6, 11, 3, 16, 5
Offset: 1

Views

Author

Emeric Deutsch, Nov 24 2011

Keywords

Comments

This is not the pruning operation mentioned in the Balaban reference (p. 360) and in the Todeschini-Consonni reference (p. 42) since in the case that the root has degree 1, this root and the incident edge are not deleted.

Examples

			a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the leaves and their incident edges, we obtain the 1-edge tree having Matula-Goebel number 2.

References

  • A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
  • R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Links

Crossrefs

Cf. A198329.
Cf. A049084, A020639, A000040.

Programs

  • Haskell
    import Data.List (genericIndex)
a198328 n = genericIndex a198328_list (n - 1)
a198328_list = 1 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a000040 (a198328 t) else a198328 r * a198328 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013
  • Maple
    with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(a(pi(n))) else a(r(n))*a(s(n)) end if end proc; seq(a(n), n = 1 .. 120);
  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = If[PrimeQ[n], Prime[a[PrimePi[n]]], Times @@ (a[#[[1]]]^#[[2]]& /@ FactorInteger[n])];
Array[a, 100] (* Jean-François Alcover, Dec 18 2017 *)

Formula

a(1)=1; a(2)=1; if n=prime(t) (t>1), then a(n)=prime(a(t)); if n=r*s (r,s,>=2), then a(n)=a(r)*a(s). The Maple program is based on this recursive formula.
Completely multiplicative with a(2) = 1, a(prime(t)) = prime(a(t)) for t > 1. - Andrew Howroyd, Aug 01 2018

A198333 Irregular triangle read by rows: row n is the pruning partition of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 1, 4, 2, 3, 1, 2, 3, 1, 4, 1, 2, 3, 1, 2, 3, 2, 2, 2, 1, 2, 3, 2, 4, 1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 4, 1, 2, 3, 2, 2, 3, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 3, 2
Offset: 1

Views

Author

Emeric Deutsch, Nov 25 2011

Keywords

Comments

The pruning partition of a tree is the reverse sequence of the number of vertices of degree 1, deleted at the successive prunings. By pruning we mean the deletion of vertices of degree 1 and of their incident edges. See the Balaban reference (p. 360) and/or the Todeschini-Consonni reference (p. 42).
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

Examples

			Row 7 is 1,3 because the rooted tree with Matula-Goebel number 7 is Y, having 3 vertices of degree 1 and after the first pruning we obtain the 1-vertex tree.
The triangle starts:  |   Squared    |  Sum of squares (= A198334(n)).
1;                           1;             1
2;                           4;             4
1,2;                         1,4;           5
1,2;                         1,4;           5
2,2;                         4,4;           8
2,2;                         4,4;           8
1,3;                         1,9;          10
1,3;                         1,9;          10
1,2,2;                       1,4,4;         9
- edited by _Antti Karttunen_, Mar 07 2017

References

  • A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
  • R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Links

Crossrefs

Cf. A061775, A198329, A198334.

Programs

  • Maple
    with(numtheory): N := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc; if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: MS := proc (m) local A, i: A[m, 1] := m: for i from 2 while 1 < A[m, i-1] do A[m, i] := a(A[m, i-1]) end do: if A[m, i-2] = 2 then [seq(A[m, j], j = 1 .. i-2)] else [seq(A[m, j], j = 1 .. i-1)] end if end proc: PP := proc (n) local NVP, q: q := nops(MS(n)): NVP := map(N, MS(n)): NVP[q], seq(NVP[q-j]-NVP[q-j+1], j = 1 .. nops(NVP)-1) end proc: for n to 21 do PP(n) end do; # for the rooted tree with Matula-Goebel number n, N(n)=A061775(n) is the number of vertices, a(n) (=A198329(n)) is the Matula-Goebel number of the tree obtained after one pruning, MS(n) is the sequence of Matula-Goebel numbers of the trees obtained after 0,1,2,... prunings, PP(n) is the pruning partition, i.e. the number of vertices of degree 1 deleted at the successive prunings, given in reverse order.

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the Matula-Goebel numbers of the trees obtained by successive prunings. Finding the number of vertices of these trees and taking differences lead to the pruning partition (see the Maple program and the explanation given there).

A198330 Matula-Goebel numbers of central rooted trees.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 10, 11, 16, 18, 19, 20, 21, 23, 25, 26, 27, 29, 32, 33, 34, 36, 40, 41, 42, 46, 49, 50, 52, 53, 54, 57, 59, 61, 62, 63, 64, 65, 66, 68, 71, 72, 73, 74, 75, 77, 80, 81, 83, 84, 85, 86, 87, 92, 97, 98, 99, 100, 101, 103, 104, 108, 114, 115
Offset: 1

Views

Author

Emeric Deutsch, Nov 24 2011

Keywords

Comments

A tree is said to be central if its center consists of one point (see the Harary reference, p. 35).

Examples

			7 is in the sequence because the rooted tree with Matula-Goebel number 7 is Y which is central. 5 is not in the sequence because the corresponding rooted tree is the path-tree on 4 vertices, a bicentral tree.

References

  • F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
  • I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
  • I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
  • D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
  • A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
  • J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).

Links

Crossrefs

Cf. A198329, A198331.

Programs

  • Maple
    with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc; b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: C := {}: for n to 130 do B := {n}: nn := n: for i while a(nn) > 1 do B := `union`(B, {a(nn)}): nn := a(nn) end do: if member(2, B) = false then C := `union`(C, {n}) else  end if end do: C;

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will not lead to the Matula-Goebel number 2 (corresponding to the 1-edge tree), then the starting rooted tree is central. The Maple program is based on this.

A198331 Matula-Goebel numbers of bicentral rooted trees.

Original entry on oeis.org

2, 5, 6, 12, 13, 14, 15, 17, 22, 24, 28, 30, 31, 35, 37, 38, 39, 43, 44, 45, 47, 48, 51, 55, 56, 58, 60, 67, 69, 70, 76, 78, 79, 82, 88, 89, 90, 91, 93, 94, 95, 96, 102, 105, 106, 107, 109, 110, 111, 112, 113, 116, 117, 118, 119, 120, 129, 135, 138, 140, 142
Offset: 1

Views

Author

Emeric Deutsch, Nov 24 2011

Keywords

Comments

A tree is said to be bicentral if its center consists of two points (see the Harary reference, p. 35).

Examples

			5 is in the sequence because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices which is bicentral.

References

  • F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
  • I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
  • I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
  • D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
  • A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
  • J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).

Links

Crossrefs

Cf. A198329, A198330.

Programs

  • Maple
    with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: CC := {}: for n to 180 do B := {n}: nn := n: for i while a(nn) > 1 do B := `union`(B, {a(nn)}): nn := a(nn) end do: if member(2, B) = true then CC := `union`(CC, {n}) else  end if end do: CC;

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will lead to the Matula-Goebel number 2 (corresponding to the 1-edge tree), then the starting rooted tree is central. The Maple program is based on this.

A198336 Irregular triangle read by rows: row n is the sequence of Matula-Goebel numbers of the rooted trees obtained from the rooted tree with Matula-Goebel number n by pruning it successively 0,1,2,... times. The operation of pruning consists of the removal of the vertices of degree one, together with their incident vertices.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 2, 1, 6, 2, 1, 7, 1, 8, 1, 9, 4, 1, 10, 3, 1, 11, 3, 1, 12, 2, 1, 13, 2, 1, 14, 2, 1, 15, 6, 2, 1, 16, 1, 17, 2, 1, 18, 4, 1, 19, 1, 20, 3, 1, 21, 4, 1, 22, 5, 2, 1, 23, 4, 1, 24, 2, 1, 25, 9, 4, 1, 26, 3, 1, 27, 8, 1, 28, 2, 1, 29, 3, 1
Offset: 1

Views

Author

Emeric Deutsch, Dec 01 2011

Keywords

Comments

This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
First entry of row n is n, last entry is 1.
Number of entries in row n is 1 + the radius of the corresponding rooted tree.

Examples

			Row 7 is 7, 1 because the rooted tree with Matula-Goebel number 7 is Y and after the first pruning we obtain the 1-vertex tree having Matula-Goebel number 1. Row 5 is 5, 2, 1 because it refers to the path tree on four vertices; after pruning it becomes the 1-edge tree with Matula-Goebel number 2.
Triangle starts:
1;
2,1;
3,1;
4,1;
5,2,1;
6,2,1;
7,1;
8,1;
9,4,1;
10,3,1;

References

  • A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
  • F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
  • I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
  • I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
  • D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
  • R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Crossrefs

Cf. A198329.

Programs

  • Maple
    with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: S := proc (m) local A, i: A[m, 1] := m: for i while a(A[m, i]) < A[m, i] do A[m, i+1] := a(A[m, i]) end do: seq(A[m, j], j = 1 .. i) end proc: for n to 15 do S(n) end do;

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the Matula-Goebel numbers of the trees obtained by successive prunings. The Maple program is based on this.
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