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A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2*a(n))+1, a(A091242(n)) = 2*a(n), 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).

%I A246201 #16 Aug 20 2014 23:59:48
%S A246201 1,3,7,2,6,14,15,4,12,28,5,30,13,8,24,56,10,60,29,26,16,48,112,20,31,
%T A246201 120,58,52,32,96,9,224,40,62,240,116,25,104,64,192,57,18,448,80,124,
%U A246201 480,11,232,50,208,128,384,114,36,61,896,160,248,27,960,17,22,464,100,416,256,49,768,228,72,122,1792,113,320,496,54,1920,34,44
%N A246201 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = (2*a(n))+1, a(A091242(n)) = 2*a(n), 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).
%C A246201 Because 2 is the only even term in A014580, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
%C A246201 Note that for any value k in A246156, "Odd reducible polynomials over GF(2)": 5, 9, 15, 17, 21, 23, ..., a(k) will be even, and apart from 2, all other even numbers are mapped to some even number, so all those terms reside in infinite cycles. Furthermore, apart from 5 and 15, all of them reside in separate cycles. The infinite cycle containing 5 and 15 goes as: ..., 47, 11, 5, 6, 14, 8, 4, 2, 3, 7, 15, 24, 20, 26, 120, 7680, ... and it is only because a(2) = 3, that it can temporarily switch back from even terms to odd terms, until after a(15) = 24 it is finally doomed to the eternal evenness.
%C A246201 (Compare also to the comments given at A246161).
%H A246201 Antti Karttunen, <a href="/A246201/b246201.txt">Table of n, a(n) for n = 1..10001</a>
%H A246201 <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%H A246201 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A246201 a(1) = 1, and for n > 1, if A091225(n) = 1 [i.e. when n is in A014580], a(n) = 1 + (2*a(A091226(n))), otherwise a(n) = 2*a(A091245(n)).
%F A246201 As a composition of related permutations:
%F A246201 a(n) = A054429(A245701(n)).
%F A246201 a(n) = A006068(A246161(n)).
%F A246201 a(n) = A193231(A246163(n)).
%F A246201 a(n) = A246203(A193231(n)).
%F A246201 Other identities:
%F A246201 For all n > 1, A000035(a(n)) = A091225(n). [After 1 maps binary representations of reducible GF(2) polynomials to even numbers and the corresponding representations of irreducible polynomials to odd numbers, in some order. A246203 has the same property].
%o A246201 (Scheme, with memoization-macro definec)
%o A246201 (definec (A246201 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (+ 1 (* 2 (A246201 (A091226 n))))) (else (* 2 (A246201 (A091245 n))))))
%Y A246201 Inverse: A246202.
%Y A246201 Similar or related permutations: A245701, A246161, A006068, A054429, A193231, A246163, A246203, A237427.
%Y A246201 Cf. A091225, A091226, A091245.
%K A246201 nonn
%O A246201 1,2
%A A246201 _Antti Karttunen_, Aug 19 2014