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%I #24 Dec 10 2019 12:09:47
%S 1,2,4,3,8,9,6,5,16,21,18,25,12,15,10,7,32,45,42,55,36,51,50,49,24,33,
%T 30,35,20,27,14,11,64,93,90,115,84,123,110,91,72,105,102,125,100,147,
%U 98,121,48,69,66,85,60,87,70,77,40,57,54,65,28,39,22,13,128,189,186,235,180,267,230,203,168,249,246,305,220,327,182,187,144
%N Tree of Eratosthenes, mirrored: a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).
%C This sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent, and each child to the right is obtained by applying A250469 to the parent:
%C 1
%C |
%C ...................2...................
%C 4 3
%C 8......../ \........9 6......../ \........5
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C / \ / \ / \ / \
%C 16 21 18 25 12 15 10 7
%C 32 45 42 55 36 51 50 49 24 33 30 35 20 27 14 11
%C etc.
%C Sequence A252753 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.
%H Antti Karttunen, <a href="/A252755/b252755.txt">Table of n, a(n) for n = 0..8192</a>
%H Antti Karttunen, <a href="/A135141/a135141.pdf">Entanglement Permutations</a>, 2016-2017
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(0) = 1, a(1) = 2; after which, a(2n) = 2*a(n), a(2n+1) = A250469(a(n)).
%F As a composition of related permutations:
%F a(n) = A252753(A054429(n)).
%F a(n) = A250245(A163511(n)).
%t (* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1 + 2 == k2, Return[m2]]]];
%t a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], 2 a[n/2], b[a[(n-1)/2]]];
%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Mar 08 2016 *)
%o (Scheme, with memoization-macro definec)
%o (definec (A252755 n) (cond ((<= n 1) (+ n 1)) ((even? n) (* 2 (A252755 (/ n 2)))) (else (A250469 (A252755 (/ (- n 1) 2))))))
%Y Inverse: A252756.
%Y Row sums: A253787, products: A253788.
%Y Similar permutations: A163511, A252753, A054429, A163511, A250245, A269865.
%Y Cf. also: A249814 (Compare the scatterplots).
%Y Cf. A083221, A250469, A253555.
%K nonn,tabf,look
%O 0,2
%A _Antti Karttunen_, Jan 02 2015
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