A106447 Doubly-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091205 and A106445.

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 23, 10, 27, 16, 81, 30, 13, 36, 25, 14, 69, 24, 11, 46, 45, 20, 21, 54, 19, 512, 57, 162, 115, 60, 47, 26, 63, 72, 61, 50, 33, 28, 135, 138, 17, 48, 35, 22, 19683, 92, 39, 90, 37, 40, 207, 42, 83, 108, 29, 38, 75, 64, 225, 114

Offset

0,3

Comments

Differs from A091205 for the first time at n=32, where A091205(32)=32, while a(32)=512. Differs from A106445 for the first time at n=13, where A106445(13)=11, while a(13)=23.

Links

Table of n, a(n) for n=0..66.

A. Karttunen, Scheme-program for computing this sequence.

Index entries for sequences that are permutations of the natural numbers

Formula

a(0)=0, a(1)=1. For irreducible GF(2)[X] polynomials ir_i with index i (i.e. A014580(i)), a(ir_i) = A000040(a(i)) and for composite polynomials n = A048723(ir_i, e_i) X A048723(ir_j, e_j) X A048723(ir_k, e_k) X ..., a(n) = a(ir_i)^a(e_i) * a(ir_j)^a(e_j) * a(ir_k)^a(e_k) * ... = A000040(a(i))^a(e_i) * A000040(a(j))^a(e_j) * A000040(a(k))^a(e_k), where X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and A048723(n, y) raises the n-th GF(2)[X] polynomial to the y:th power, while * is the ordinary multiplication and ^ is the ordinary exponentiation.

Example

a(5) = 9, as 5 encodes the GF(2)[X] polynomial x^2+1, which is the square of the second irreducible GF(2)[X] polynomial x+1 (encoded as 3), a(2)=2 and the square of the second prime is 3^2=9. a(13) = a(A014580(5)) = A000040(a(5)) = A000040(9) = 23. a(32) = a(A048723(2,5)) = a(2)^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = a(3) * a(2)^a(4) = 3 * 2^4 = 3 * 16 = 48.

Crossrefs

Inverse: A106446. Variant: A091205.

Sequence in context: A106445 A106443 A091205 * A356886 A359416 A222248

Adjacent sequences: A106444 A106445 A106446 * A106448 A106449 A106450

Keyword

nonn

Author

Antti Karttunen, May 09 2005

Status

approved