A106443 Exponent-recursed cross-domain bijection from GF(2)[X] to N. Position of A106456(n) in A075166.

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, 768, 35, 34, 19683, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47

Offset

0,3

Comments

This map from the multiplicative domain of GF(2)[X] to that of N preserves Catalan-family structures, e.g. A075164(n) = a(A106454(n)), A106453(n) = A075163(a(n)), A106455(n) = A075165(a(n)), A106456(n) = A075166(a(n)), A106457(n) = A075167(a(n)). Shares with A091203 and A106445 the property that maps A014580(n) to A000040(n). Differs from the former for the first time at n=32, where A091203(32)=32, while a(32)=512. Differs from the latter for the first time at n=48, where A106445(48)=48, while a(48)=768.

Links

Table of n, a(n) for n=0..67.

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(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(1+e_j)-1) * a(ir_k)^(a(1+e_k)-1) * ... = A000040(i)^a(e_i) * A000040(j)^(a(1+e_j)-1) * A000040(k)^(a(1+e_k)-1), 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. Here ir_i is the most significant (largest) irreducible polynomial in the factorization of n; its exponent e_i is not incremented before the recursion step, while the exponents of less significant factors e_j, e_k, ... are incremented by one before recursing and the result of the recursion is decremented by one before use.

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) 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+1)-1) = 3 * 2^(9-1) = 3 * 256 = 768.

Crossrefs

Inverse: A106442. a(n) = A075164(A106453(n)).

Sequence in context: A260741 A091203 A106445 * A091205 A106447 A356886

Adjacent sequences: A106440 A106441 A106442 * A106444 A106445 A106446

Keyword

nonn

Author

Antti Karttunen, May 09 2005

Status

approved