This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A286105 #19 May 07 2021 12:34:38 %S A286105 0,1,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,5,5,5,6,6,6,6,6,6,6,6,6,6,7,6, %T A286105 6,7,6,7,6,7,7,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,7,8,8,7, %U A286105 7,7,7,7,8,7,8,8,8,7,8,8,7,8,8,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8 %N A286105 a(1) = 0; for n > 1, a(n) = 1 + max(a(A285734(n)), a(A285735(n))). %C A286105 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. %H A286105 Antti Karttunen, <a href="/A286105/b286105.txt">Table of n, a(n) for n = 1..10000</a> %F A286105 a(1) = 0 and for n > 1, a(n) = 1 + max(a(A285734(n)), a(A285735(n))). %F A286105 a(1) = 1 and for n > 1, a(n) = 1 + a(A286107(n)). %F A286105 Other identities. For all n >= 1: %F A286105 a(2*A005117(n)) = 1+a(A005117(n)). %e A286105 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. %e A286105 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. %e A286105 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. %e A286105 The tree with root 17 looks like this: %e A286105 17 %e A286105 | %e A286105 ..................../ \.................. %e A286105 7 10 %e A286105 2......../ \........5 5......../ \........5 %e A286105 / \ / \ / \ / \ %e A286105 / \ / \ / \ / \ %e A286105 / \ / \ / \ / \ %e A286105 1 1 2 3 2 3 2 3 %e A286105 1 1 1 2 1 1 1 2 1 1 1 2 %e A286105 1 1 1 1 1 1 %e A286105 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. %o A286105 (Scheme, with memoization-macro definec) %o A286105 (definec (A286105 n) (if (= 1 n) 0 (+ 1 (max (A286105 (A285734 n)) (A286105 (A285735 n)))))) %o A286105 (definec (A286105 n) (if (= 1 n) 0 (+ 1 (A286105 (A286107 n))))) %o A286105 (Python) %o A286105 from sympy.ntheory.factor_ import core %o A286105 def issquarefree(n): return core(n) == n %o A286105 def a285734(n): %o A286105 if n==1: return 0 %o A286105 j=n//2 %o A286105 while True: %o A286105 if issquarefree(j) and issquarefree(n - j): return j %o A286105 else: j-=1 %o A286105 def a285735(n): return n - a285734(n) %o A286105 def a286105(n): return 0 if n==1 else 1 + max(a286105(a285734(n)), a286105(a285735(n))) %o A286105 print([a286105(n) for n in range(1, 121)]) # _Indranil Ghosh_, May 02 2017 %Y A286105 Cf. A005117, A285734, A285735, A286103, A286104, A286106, A286107. %K A286105 nonn %O A286105 1,3 %A A286105 _Antti Karttunen_, May 02 2017