A005517 Smallest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.

1, 2, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 625, 1125, 1875, 3125, 5625, 9375, 15625, 28125, 46875, 78125, 140625, 234375, 390625, 703125, 1171875, 1953125, 3515625, 5859375, 9765625, 17578125, 29296875, 48828125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.

For n >= 3, this is also the minimum number of Hamiltonian paths in a strong tournament with n vertices (Busch). - Gordon Royle, Jan 24 2022

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Arthur H. Busch, A Note on the Number of Hamiltonian Paths in Strong Tournaments, Electronic Journal of Combinatorics, N3 (2006).
• 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, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., 107, No. 19, 1994, 65-74.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for sequences related to Matula-Goebel numbers
• Index entries for sequences related to rooted trees
• Index entries for sequences related to trees
• Index entries for linear recurrences with constant coefficients, signature (0,0,5).

Crossrefs

Cf. A061773. See A005518 for the largest value of f(T).

Programs

Maple

a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif `mod`(n, 3) = 0 then 3*5^((1/3)*n-1) elif `mod`(n, 3) = 1 then 5^((1/3)*n-1/3) else 9*5^((1/3)*n-5/3) end if end proc: seq(a(n), n = 1 .. 34); # Emeric Deutsch, Apr 15 2012
A005517:=(-1-2*z-3*z**2+z**4)/(-1+5*z**3); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Join[{1,2},LinearRecurrence[{0,0,5},{3,5,9},40]] (* Harvey P. Dale, Feb 25 2012 *)
a[n_] := Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^((1/3)*n-1), Mod[n, 3] == 1, 5^((1/3)*n-1/3), True, 9*5^((1/3)*n-5/3)]; Table[a[n], {n, 1, 34}] (* Jean-François Alcover, Mar 06 2014, after Emeric Deutsch *)

Formula

a(1)=1; a(2)=2; a(n) = 3*5^((n-3)/3) if n=0 (mod 3); a(n)=5^((n-1)/3) if n>=4 and n=1 (mod 3); a(n)=9*5^((n-5)/3) if n>=5 and n = 2 (mod 3) (see the Gutman and Ivic 1994 paper). - Emeric Deutsch, Apr 15 2012

G.f.: z*(1+2*z+3*z^2-z^4)/(1-5*z^3) (conjectured by Simon Plouffe).

a(n+3) = 5*a(n) for n >= 3 under plausible assumptions about growth of prime numbers. - David W. Wilson, Jul 05 2001

A091233(n) = (A005518(n)-a(n))+1. - Antti Karttunen, May 24 2004