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

Author

Antti Karttunen, May 09 2005

Keywords

• nonn

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.

Examples

			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.
		

Links

• A. Karttunen, Scheme-program for computing this sequence.
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A106446. Variant: A091205.

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.