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%I #45 Oct 13 2022 05:56:14
%S 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,
%T 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,
%U 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
%N a(n) is the recursion depth of repeatedly factoring n and then the exponents in its prime product factorization, until 1 is reached.
%C 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.
%C 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.
%C 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.
%H Charles R Greathouse IV, <a href="/A185102/b185102.txt">Table of n, a(n) for n = 1..10000</a>
%H Vitalii V. Iudelevich, <a href="https://arxiv.org/abs/2210.06055">On The Tree Structure of Natural Numbers, II</a>, arXiv:2210.06055 [math.NT], 2022. See Figure 2 p. 2.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetration">Tetration</a>
%F a(1) = 0. a(p1^k1*p2^k2*...*pn^kn) = 1 + max{a(k1), a(k2), ..., a(kn)} with all pi prime and distinct.
%F 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
%e a(156) = a(2^2*3*13) = 1 + max{a(2), a(1)} = 1 + max{1, 0} = 1 + 1 = 2
%t 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 *)
%t a = {0}; Do[AppendTo[a, 1 + Max @@ a[[Union[FactorInteger[n][[All, 2]]]]]], {n, 2, 105}]; a (* _Ray Chandler_, Nov 22 2013 *)
%o (PARI) a(n)=if(n<4,n>1,vecmax(apply(a,Set(factor(n)[,2])))+1) \\ _Charles R Greathouse IV_, Nov 21 2013
%Y Cf. A096309.
%K nonn
%O 1,4
%A _Colm Bhandal_, Jan 23 2012
%E Corrected (n=32, 36, 64, 72, 96, 100) by _Jean-François Alcover_, Nov 20 2013
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