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A257538
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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.
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3
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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
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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.
(End)
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := a[n] = Which[n == 1, 1, PrimeOmega[n] == 1, Prime[ Prime[ a[PrimePi[n]]]], True, a[r[n]]*a[s[n]]];
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PROG
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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)
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CROSSREFS
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KEYWORD
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nonn,mult,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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