%I #27 Dec 10 2019 12:09:40

%S 1,2,4,3,5,9,8,7,11,19,6,17,10,15,23,39,13,35,18,21,31,47,79,27,16,71,

%T 37,43,63,95,14,159,55,33,143,75,22,87,127,191,38,29,319,111,67,287,

%U 12,151,45,175,255,383,77,59,34,639,223,135,20,575,30,25,303,91,351,511,46,767,155,119,69,1279,78,447,271,41,1151,61,51

%N Permutation of natural numbers: a(1) = 1, a(A014580(n)) = 2*a(n), a(A091242(n)) = 2*a(n)+1, where A014580(n) = binary code for n-th irreducible polynomial over GF(2), A091242(n) = binary code for n-th reducible polynomial over GF(2).

%H Antti Karttunen, <a href="/A245701/b245701.txt">Table of n, a(n) for n = 1..10001</a>

%H Antti Karttunen, <a href="/A135141/a135141.pdf">Entanglement Permutations</a>, 2016-2017

%H <a href="/index/Ge#GF2X">Index entries for sequences related to polynomials in ring GF(2)[X]</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(1) = 1, and for n > 1, if n is in A014580, a(n) = 2*a(A091226(n)), otherwise a(n) = 1 + 2*a(A091245(n)).

%F As a composition of related permutations:

%F a(n) = A135141(A245704(n)).

%F Other identities:

%F For all n >= 1, 1 - A000035(a(n)) = A091225(n). [Maps binary representations of irreducible GF(2) polynomials (= A014580) to even numbers and the corresponding representations of reducible polynomials to odd numbers].

%o (PARI)

%o allocatemem(123456789);

%o A091226 = vector(2^22);

%o isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_

%o n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)

%o A091245(n) = ((n-A091226[n])-1);

%o A245701(n) = if(1==n, 1, if(isA014580(n), 2*(A245701(A091226[n])), 1 + 2*(A245701(A091245(n)))));

%o for(n=1, 10001, write("b245701.txt", n, " ", A245701(n)));

%o (Scheme, with memoizing definec-macro)

%o (definec (A245701 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (* 2 (A245701 (A091226 n)))) (else (+ 1 (* 2 (A245701 (A091245 n)))))))

%Y Inverse: A245702.

%Y Cf. A000035, A014580, A091242, A091225, A091226, A091245.

%Y Similar entanglement permutations: A135141, A193231, A237427, A243287, A245703, A245704.

%K nonn,look,hear

%O 1,2

%A _Antti Karttunen_, Aug 02 2014