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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.

%I A257538 #33 Sep 09 2024 11:55:40
%S A257538 1,3,11,9,127,33,83,27,121,381,5381,99,773,249,1397,81,3001,363,563,
%T A257538 1143,913,16143,4943,297,16129,2319,1331,747,23563,4191,648391,243,
%U A257538 59191,9003,10541,1089,3761,1689,8503,3429,57943,2739,13297,48429
%N 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.
%C A257538 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.
%C A257538 Fully multiplicative with a(prime(n)) = prime(prime(a(n))). - _Antti Karttunen_, Mar 09 2017
%H A257538 Antti Karttunen, <a href="/A257538/b257538.txt">Table of n, a(n) for n = 1..708</a>
%H A257538 Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H A257538 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A257538 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A257538 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A257538 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A257538 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A257538 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A257538 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.
%F A257538 From _Antti Karttunen_, Mar 09 2017: (Start)
%F A257538 a(1) = 1; for n>1, a(n) = A000040(A000040(a(A055396(n)))) * a(A032742(n)).
%F A257538 A046523(a(n)) = A046523(n). [Preserves the prime-signature of n].
%F A257538 (End)
%e A257538 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.
%p A257538 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);
%t A257538 r[n_] := FactorInteger[n][[1, 1]];
%t A257538 s[n_] := n/r[n];
%t A257538 a[n_] := a[n] = Which[n == 1, 1, PrimeOmega[n] == 1, Prime[ Prime[ a[PrimePi[n]]]], True, a[r[n]]*a[s[n]]];
%t A257538 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Sep 09 2024, after Maple program *)
%o A257538 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
%o A257538 (Scheme, with memoization-macro definec)
%o A257538 (definec (A257538 n) (cond ((= 1 n) 1) (else (* (A000040 (A000040 (A257538 (A055396 n)))) (A257538 (A032742 n))))))
%o A257538 ;; _Antti Karttunen_, Mar 09 2017
%Y A257538 Cf. A000040, A000720, A006450, A032742, A046523, A049084, A055396.
%K A257538 nonn,mult
%O A257538 1,2
%A A257538 _Emeric Deutsch_, May 01 2015
%E A257538 Formula corrected by _Antti Karttunen_, Mar 09 2017