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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

• 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.
• Index entries for sequences related to Matula-Goebel numbers.
• Index entries for sequences related to rooted trees.
• Index entries for sequences related to trees.

Crossrefs

Apart from initial terms, same as A057452.

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

Programs

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 *)

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

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