A106445 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Variant of A091203 and A106443.

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 512, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47

Offset: 0

Author

Antti Karttunen, May 09 2005

Keywords

• nonn

Comments

This map from the multiplicative domain of GF(2)[X] to that of N preserves 'superfactorized' structures, e.g. A106493(n) = A106490(a(n)), A106494(n) = A106491(a(n)), A106495(n) = A064372(a(n)). Shares with A091203 and A106443 the property that maps A014580(n) to A000040(n). Differs from the plain variant A091203 for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the variant A106443 for the first time at n=48, where A106443(48)=768, while a(48)=48. Differs from a yet deeper variant A106447 for the first time at n=13, where A106447(13)=23, while a(13)=11.

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) and the square of the second prime is 3^2=9. a(32) = a(A048723(2,5)) = 2^a(5) = 2^9 = 512. a(48) = a(3 X A048723(2,4)) = 3 * 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: A106444.

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(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(i)^a(e_i) * A000040(j)^a(e_j) * A000040(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.