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A369015
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Matula-Goebel number of the prime tower factorization tree of n.
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6
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1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4
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OFFSET
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1,2
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COMMENTS
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The prime tower factorization tree of n having prime factorization n = Product p_i^e_i comprises a root vertex and beneath it child subtrees with tree numbers e_i.
The Matula-Goebel number represents a rooted tree (no ordering among siblings), so the primes p_i have no effect, just the exponents.
Runs of various consecutive equal values occur (so the same tree structure), and n = A368899(k) is the first place where a run of length >= k begins.
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LINKS
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FORMULA
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a(n) = Product prime(a(e_i)) where e_i = A124010(n,i) is each exponent in the prime factorization of n.
Multiplicative with a(p^e) = prime(a(e)) for prime p.
a(n) = 2^k if and only if n is the product of k distinct primes.
a(n) = 3 if and only if n is a prime power of a prime number (A053810).
(End)
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EXAMPLE
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n = 274274771783272 = 2^3 * 13^(3^2) * 53^1 * 61^1 has exponents 3, 9, 1, 1 which become the following prime tower factorization tree, and corresponding Matula-Goebel number a(n) = 60:
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n=274274771783272 a(n)=60
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3 9 1 1 2 3 1 1
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1 2 1 2
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1 1
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PROG
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(PARI) a(n) = vecprod([prime(self()(e)) |e<-factor(n)[, 2]]);
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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