A252753 - OEIS

A252753

Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

%I #30 Mar 08 2016 05:13:39

%S 1,2,3,4,5,6,9,8,7,10,15,12,25,18,21,16,11,14,27,20,35,30,33,24,49,50,

%T 51,36,55,42,45,32,13,22,39,28,65,54,57,40,77,70,87,60,85,66,69,48,

%U 121,98,147,100,125,102,105,72,91,110,123,84,115,90,93,64,17,26,63,44,95,78,81,56,119,130,159,108,145,114,117,80

%N Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

%C This sequence can be represented as a binary tree. Each child to the left is obtained by applying A250469 to the parent, and each child to the right is obtained by doubling the parent:

%C 1

%C |

%C ...................2...................

%C 3 4

%C 5......../ \........6 9......../ \........8

%C / \ / \ / \ / \

%C 7 10 15 12 25 18 21 16

%C 11 14 27 20 35 30 33 24 49 50 51 36 55 42 45 32

%C etc.

%C Sequence A252755 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="/A252753/b252753.txt">Table of n, a(n) for n = 0..8192</a>

%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) = A250469(a(n)), a(2n+1) = 2 * a(n).

%F As a composition of related permutations:

%F a(n) = A252755(A054429(n)).

%F a(n) = A250245(A005940(1+n)).

%F Other identities. For all n >= 1:

%F A055396(a(n)) = A001511(n). [A005940 has the same property.]

%F a(A003945(n)) = A001248(n) for n>=1. - _Peter Luschny_, Jan 13 2015

%t (* b = A250469 *)

%t 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], b[a[n/2]], 2 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 (A252753 n) (cond ((<= n 2) (+ 1 n)) ((even? n) (A250469 (A252753 (/ n 2)))) (else (* 2 (A252753 (/ (- n 1) 2))))))

%Y Inverse: A252754.

%Y Row sums: A253787, products: A253788.

%Y Fixed points of a(n-1): A253789.

%Y Similar permutations: A005940, A252755, A054429, A250245.

%Y Cf. also A001248, A001511, A055396, A083221, A181565, A250469, A253555.

%K nonn,tabf,nice

%O 0,2

%A _Antti Karttunen_, Jan 02 2015