A252753 Tree of Eratosthenes: a(0) = 1, a(1) = 2; after which, a(2n) = A250469(a(n)), a(2n+1) = 2 * a(n).

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

Offset: 0

Author

Antti Karttunen, Jan 02 2015

Keywords

• nonn
• tabf
• nice

Comments

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:

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 51 36 55 42 45 32

etc.

Sequence A252755 is the mirror image of the same tree. A253555(n) gives the distance of n from 1 in both trees.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A252754.

Row sums: A253787, products: A253788.

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

Similar permutations: A005940, A252755, A054429, A250245.

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

Programs

Mathematica

(* 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]]]];
a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ[n], b[a[n/2]], 2 a[(n-1)/2]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2016 *)

Formula

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

As a composition of related permutations:

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

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

Other identities. For all n >= 1:

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

a(A003945(n)) = A001248(n) for n>=1. - Peter Luschny, Jan 13 2015