

A235042


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


18



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
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OFFSET

0,3


COMMENTS

Like A091203 this is a factorizationpreserving 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 setwise difference A014580 \ A000040 (= A091214) in numerical order to the setwise 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.


LINKS



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).


EXAMPLE

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.


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



