A005518 Largest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling. (Formerly M1154)

1, 2, 4, 8, 19, 67, 331, 2221, 19577, 219613, 3042161, 50728129, 997525853, 22742734291, 592821132889, 17461204521323, 575411103069067, 21034688742654437, 846729487306354343, 37291202558050631867, 1786728150827024765159, 92665443023071791629459

Offset

1,2

Comments

Let prime(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 prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.

References

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

Links

Table of n, a(n) for n=1..22.

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B, Vol. 29, No. 1 (1980), pp. 141-143.

I. Gutman and A. Ivić, Graphs with maximal and minimal Matula numbers, Bulletin CVII Acad. Serbe, Sciences Math., Vol. 107, No. 19 (1994), pp. 65-74.

I. Gutman and A. Ivić, On Matula numbers, Discrete Math., Vol. 150, No. 1-3 (1996), pp. 131-142.

D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev., Vol. 10 (1968), p. 273.

K. Walisch, primecount: fast prime counting function library.

Index entries for sequences related to Matula-Goebel numbers.

Index entries for sequences related to rooted trees.

Index entries for sequences related to trees.

Formula

a(1)=1; a(2)=2; a(3)=4; a(4)=8; a(n) = the a(n-1)-th prime (see the Gutman and Ivic 1994 paper). - Emeric Deutsch, Apr 15 2012

Under plausible assumptions about the growth of the primes, for n >= 4, a(n+1) = a(n)-th prime and A005518(n) = A057452(n-3). - David W. Wilson, Jul 09 2001

A091233(n) = (a(n)-A005517(n))+1. - Antti Karttunen, May 24 2004

Maple

with(numtheory): a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif n = 3 then 4 elif n = 4 then 8 else ithprime(a(n-1)) end if end proc: seq(a(n), n = 1 .. 12); # Emeric Deutsch, Apr 15 2012

Mathematica

a[n_] := a[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, _, Prime[a[n-1]]]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Mar 06 2014, after Emeric Deutsch *)

Crossrefs

Apart from initial terms, same as A057452.

Cf. A061773. See A005517 for the smallest value of f(T).

Sequence in context: A006897 A287025 A034767 * A326997 A014225 A217520

Adjacent sequences: A005515 A005516 A005517 * A005519 A005520 A005521

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Jul 09 2001

a(17)-a(19) from Robert G. Wilson v, Mar 07 2017 using Kim Walisch's primecount

a(20)-a(22) from Robert P. P. McKone, Jun 14 2026

Status

approved