

A257538


The Matula number of the rooted tree obtained from the rooted tree T having Matula number n by replacing each edge of T with a path of length 2.


3



1, 3, 11, 9, 127, 33, 83, 27, 121, 381, 5381, 99, 773, 249, 1397, 81, 3001, 363, 563, 1143, 913, 16143, 4943, 297, 16129, 2319, 1331, 747, 23563, 4191, 648391, 243, 59191, 9003, 10541, 1089, 3761, 1689, 8503, 3429, 57943, 2739, 13297, 48429
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OFFSET

1,2


COMMENTS

The Matula (or MatulaGoebel) number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the Matula 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 numbers of the m branches of T.
Fully multiplicative with a(prime(n)) = prime(prime(a(n))).  Antti Karttunen, Mar 09 2017


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..708
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 23142322.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index to divisibility sequences
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let p(n) denote the nth prime (= A000040(n)). We have the recursive equations: a(p(n)) = p(p(a(n))), a(rs) = a(r)a(s), a(1) = 1. The Maple program is based on this.
From Antti Karttunen, Mar 09 2017: (Start)
a(1) = 1; for n>1, a(n) = A000040(A000040(a(A055396(n)))) * a(A032742(n)).
A046523(a(n)) = A046523(n). [Preserves the primesignature of n].
(End)


EXAMPLE

a(3)=11; indeed, 3 is the Matula number of the path of length 2 and 11 is the Matula number of the path of length 4.


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 bigomega(n) = 1 then ithprime(ithprime(a(pi(n)))) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 60);


PROG

A257538(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(prime(A257538(primepi(f[i, 1]))))); factorback(f); }; \\ Nonmemoized implementation by Antti Karttunen, Mar 09 2017
(Scheme, with memoizationmacro definec)
(definec (A257538 n) (cond ((= 1 n) 1) (else (* (A000040 (A000040 (A257538 (A055396 n)))) (A257538 (A032742 n))))))
;; Antti Karttunen, Mar 09 2017


CROSSREFS

Cf. A000040, A000720, A006450, A032742, A046523, A049084, A055396.
Sequence in context: A316884 A084409 A038229 * A304052 A303122 A080351
Adjacent sequences: A257535 A257536 A257537 * A257539 A257540 A257541


KEYWORD

nonn,mult


AUTHOR

Emeric Deutsch, May 01 2015


EXTENSIONS

Formula corrected by Antti Karttunen, Mar 09 2017


STATUS

approved



