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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The Matula (or 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  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, 2314-2322. 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 Rev. 10 (1968) 273. FORMULA Let p(n) denote the n-th 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 prime-signature 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 memoization-macro 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

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Last modified January 22 16:47 EST 2019. Contains 319364 sequences. (Running on oeis4.)