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A185102
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a(n) is the recursion depth of repeatedly factoring n and then the exponents in its prime product factorization, until 1 is reached.
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7
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0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1
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OFFSET
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1,4
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COMMENTS
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a(n) is the depth of a tree whose root is n and whose construction follows the rule: The children of n are the exponents in its prime factorization. If the number is 1, the convention is taken that the prime factorization is empty, and hence the set of children of 1 is the null set. The formulaic sequence definition given uses pattern matching on n as a product of primes, which is always possible by the fundamental theorem of arithmetic. The max{...} notation refers to the maximum value of the specified set.
The sequence differs from A096309 only at a(1) = 0. The justification for this choice is that the null product of primes has 0 levels.
For all numbers i less than or equal to 2^^x, i.e., 2 tetrated to the x, a(i) < x. Tetration is the next hyperoperation after exponentiation, and can be defined: x^^0 = 1 and x^^y = x^(x^^(y - 1)). Due to this bound the sequence can be seen to grow very slowly.
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LINKS
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FORMULA
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a(1) = 0. a(p1^k1*p2^k2*...*pn^kn) = 1 + max{a(k1), a(k2), ..., a(kn)} with all pi prime and distinct.
a(n) <= lg*(n), where lg*(x) = 0 if x < 2 and lg*(x) = lg*(log_2(x)) otherwise. - Charles R Greathouse IV, Nov 21 2013
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EXAMPLE
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a(156) = a(2^2*3*13) = 1 + max{a(2), a(1)} = 1 + max{1, 0} = 1 + 1 = 2
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MATHEMATICA
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f[n_Integer] := FactorInteger[n][[All, 2]]; a[1] = 0; a[n_] := Depth[f[n] //. k_Integer /; k > 1 :> f[k]] - 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 20 2013 *)
a = {0}; Do[AppendTo[a, 1 + Max @@ a[[Union[FactorInteger[n][[All, 2]]]]]], {n, 2, 105}]; a (* Ray Chandler, Nov 22 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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