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A216200
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Number of disjoint trees that appear while iterating the sum of divisors function up to n.
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7
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1, 2, 2, 2, 3, 3, 3, 3, 4, 5, 6, 5, 5, 5, 5, 6, 7, 6, 7, 7, 8, 9, 10, 8, 9, 10, 11, 11, 12, 12, 11, 10, 11, 12, 13, 13, 14, 14, 14, 14, 15, 13, 14, 14, 15, 16, 17, 15, 16, 17, 18, 19, 20, 19, 20, 19, 19, 20, 21, 19, 20, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27
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OFFSET
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1,2
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COMMENTS
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A tree like (2, 3, 4, 7, 8) contains all numbers below 8 such that iterating the sum of divisors function to any of them, while staying below 8, will lead to 8.
Inspired by the article in link, where a (p1, p2, p3)-tree is defined with p1 the smallest number in the tree, and p2, p3, such that all sequences {sigma^i(n)} (iterations of sigma), with p1 <= n <= p2 and sigma^i(n) < p3 have nonempty intersection with {sigma^i(p1)}. For instance, 21 (p1, 200, 10^10)-trees and 64 (p1, 1000, 10^100)-trees were found.
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LINKS
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FORMULA
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It appears that a(n)/n = 0.32721... + O(1/sqrt(n)). - M. F. Hasler, Nov 19 2019
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EXAMPLE
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For n=23, there are 10 disjoint trees: (1), (2, 3, 4, 7, 8, 15), (5, 6, 11, 12), (9, 13, 14), (10, 17, 18), (16), (19, 20), (21), (22), (23). With the arrival of 24, 3 trees are united, that is those that contain 15, 14 and 23, so that there are now 8 trees.
Some further values: a(100) = 33, a(500) = 167, a(1000) = 333.
Further values: a(10^4) = 3282, a(10^5) = 32739, a(10^6) = 327165, a(10^7) = 3272134. - M. F. Hasler, Nov 19 2019
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PROG
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(PARI)
A216200_vec(N)={my(C=Map(), s, c); vector(N, n, mapisdefined(C, n)&& c+=mapget(C, n) + mapdelete(C, n); mapput(C, s=sigma(n), if(mapisdefined(C, s), mapget(C, s))+1); n-c)} \\ Use allocatemem() for N >= 10^6.
A216200(n)={my(C=Map(), s); n-sum(n=2, n, mapput(C, s=sigma(n), if(mapisdefined(C, s), mapget(C, s))+1); if(mapisdefined(C, n), mapget(C, n) + mapdelete(C, n)))} \\ (slightly faster to compute a single value)
tree(n)=[n, if(n>1, apply(self, invsigma(n)), "fixed point")] \\ to create the tree rooted in n. (End)
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CROSSREFS
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Cf. A257669, A257670: size and smallest number of subtree rooted at n.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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