A235042 Factorization-preserving bijection from GF(2)[X]-polynomials to nonnegative integers, version which fixes the elements that are irreducible in both semirings.

0, 1, 2, 3, 4, 9, 6, 7, 8, 21, 18, 11, 12, 13, 14, 27, 16, 81, 42, 19, 36, 49, 22, 39, 24, 5, 26, 63, 28, 33, 54, 31, 32, 93, 162, 91, 84, 37, 38, 99, 72, 41, 98, 15, 44, 189, 78, 47, 48, 77, 10, 243, 52, 57, 126, 17, 56, 117, 66, 59, 108, 61, 62, 147, 64, 441

Offset: 0

Author

Antti Karttunen, Jan 02 2014

Keywords

• nonn

Comments

Like A091203 this is a factorization-preserving isomorphism from GF(2)[X]-polynomials to integers. The former are encoded in the binary representation of n like this: n=11, '1011' in binary, stands for polynomial x^3+x+1, n=25, '11001' in binary, stands for polynomial x^4+x^3+1. However, this version does not map the irreducible GF(2)[X] polynomials (A014580) straight to the primes (A000040), but instead fixes the intersection of those two sets (A091206), and maps the elements in their set-wise difference A014580 \ A000040 (= A091214) in numerical order to the set-wise difference A000040 \ A014580 (= A091209).

The composite values are defined by the multiplicativity. E.g., we have a(A048724(n)) = 3*a(n) and a(A001317(n)) = A000244(n) = 3^n for all n.

This map satisfies many of the same identities as A091203, e.g., we have A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)) and A091247(n) = A066247(a(n)) for all n >= 1.

Examples

			Here (t X u) = A048720(t,u):
a(2)=2, a(3)=3 and a(7)=7, as 2, 3 and 7 are all in A091206.
a(4) = a(2 X 2) = a(2)*a(2) = 2*2 = 4.
a(5) = a(3 X 3) = a(3)*a(3) = 3*3 = 9.
a(9) = a(3 X 7) = a(3)*a(7) = 3*7 = 21.
a(10) = a(2 X 3 X 3) = a(2)*a(3)*a(3) = 2*3*3 = 18.
a(15) = a(3 X 3 X 3) = a(3)^3 = 3^3 = 27.
a(17) = a(3 X 3 X 3 X 3) = a(3)^4 = 3^4 = 81.
a(21) = a(7 X 7) = a(7)*a(7) = 7*7 = 49.
a(25) = 5, as 25 is the first term of A091214 and 5 is the first term of A091209.
a(50) = a(2 X 25) = a(2)*a(25) = 2*5 = 10.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A235041. Fixed points: A235045.

Cf. A010051, A004247, A091225, A091209, A235044.

Similar cross-multiplicative permutations: A091203, A091205, A106443, A106445, A106447.

Formula

a(0)=0, a(1)=1, a(p) = p for those irreducible GF(2)[X]-polynomials whose binary encoding is a prime (i.e., p is in A091206), and for the rest of irreducible GF(2)[X]-polynomials (those which are encoded by a composite natural number, i.e., q is in A091214), a(q) = A091209(A235044(q)), and for reducible polynomials, a(i X j X k X ...) = a(i) * a(j) * a(k) * ..., where each i, j, k, ... is in A014580, X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).