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 A286105 a(1) = 0; for n > 1, a(n) = 1 + max(a(A285734(n)), a(A285735(n))). 4
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 OFFSET 1,3 COMMENTS By invoking A285734 and A285735 recursively, any natural number n > 1 can be decomposed as a sum of successively smaller squarefree numbers, until only n instances of 1's remain. This process can be depicted as a binary tree, where 1's are leaves, and any other node n branches to the left as A285734(n) and to the right as A285735(n). This sequence gives the distance from the root of tree (n) to a leaf (1) that is furthest removed from the root. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10000 FORMULA a(1) = 0 and for n > 1, a(n) = 1 + max(a(A285734(n)), a(A285735(n))). a(1) = 1 and for n > 1, a(n) = 1 + a(A286107(n)). Other identities. For all n >= 1: a(2*A005117(n)) = 1+a(A005117(n)). EXAMPLE A285734(2) = A285735(2) = 1, thus a tree with root 2 has just two leaves 1 and 1, so the maximum distance to them is 1, thus a(2) = 1. A285734(3) = 1 and A285735(3) = 2, thus a tree with root 3 has one immediate leave 1 and the subtree 2 as its other branch, so the distance to a farthest leaf (1) is two edges, thus a(3) = 2. A285734(5) = 2 and A285735(3) = 3, thus a tree with root 5 has the subtree 2 as its other branch, and the subtree 3 as the other branch, so the maximum distance to a leaf (1) is 1 + longest distance computed for cases 2 and 3, thus a(5) = 1 + max(1,2) = 3. The tree with root 17 looks like this:                                     17                                      |                 ..................../ \..................                 7                                       10       2......../ \........5                   5......../ \........5      / \                 / \                 / \                 / \     /   \               /   \               /   \               /   \    /     \             /     \             /     \             /     \   1       1           2       3           2       3           2       3                     1   1   1   2       1   1   1   2       1   1   1   2                                1 1                 1 1                 1 1 We see that the longest distance to 1 from the root can be found for example at the right border of the tree, five edges in total, thus a(17) = 5. PROG (Scheme, with memoization-macro definec) (definec (A286105 n) (if (= 1 n) 0 (+ 1 (max (A286105 (A285734 n)) (A286105 (A285735 n)))))) (definec (A286105 n) (if (= 1 n) 0 (+ 1 (A286105 (A286107 n))))) (Python) from sympy.ntheory.factor_ import core def issquarefree(n): return core(n) == n def a285734(n):     if n==1: return 0     j=int(n/2)     while True:         if issquarefree(j) and issquarefree(n - j): return j         else: j-=1 def a285735(n): return n - a285734(n) def a286105(n): return 0 if n==1 else 1 + max(a286105(a285734(n)), a286105(a285735(n))) print [a286105(n) for n in xrange(1, 121)] # Indranil Ghosh, May 02 2017 CROSSREFS Cf. A005117, A285734, A285735, A286103, A286104, A286106, A286107. Sequence in context: A274102 A261100 A130249 * A061071 A122258 A263089 Adjacent sequences:  A286102 A286103 A286104 * A286106 A286107 A286108 KEYWORD nonn AUTHOR Antti Karttunen, May 02 2017 STATUS approved

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Last modified May 22 07:30 EDT 2019. Contains 323478 sequences. (Running on oeis4.)