A246164 - OEIS

%I #17 Aug 20 2014 23:59:02

%S 1,2,4,11,8,5,3,7,6,9,13,17,47,31,14,61,21,42,185,24,87,319,62,12,25,

%T 19,10,59,20,15,37,229,49,22,67,76,415,103,28,18,55,137,34,41,16,27,

%U 97,78,425,109,29,1627,222,54,283,433,79,373,3053,33,131,647,108,847,133,745,6943,44,193,1053,160,504,4333,587,99

%N Permutation of natural numbers: a(1) = 1, a(A065621(n)) = A014580(a(n-1)), a(A048724(n)) = A091242(a(n)), where A065621(n) and A048724(n) are the reversing binary representation of n and -n, respectively, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

%C This is an instance of entanglement permutation, where the two complementary pairs to be entangled with each other are A065621/A048724 and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).

%C The former are themselves permutations of A000069/A001969 (odious and evil numbers), which means that this permutation shares many properties with A246162.

%C For the comments about the cycle structure, please see A246163.

%H Antti Karttunen, <a href="/A246164/b246164.txt">Table of n, a(n) for n = 1..195</a>

%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</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 A010060(n) = 1 [i.e. when n is an odious number], a(n) = A014580(a(A065620(n)-1)), otherwise a(n) = A091242(a(- (A065620(n)))). [A065620 Converts sum of powers of 2 in binary representation of n to an alternating sum].

%F As a composition of related permutations:

%F a(n) = A246202(A193231(n)).

%F a(n) = A245702(A234025(n)).

%F a(n) = A246162(A234612(n)).

%F a(n) = A193231(A246204(A193231(n))).

%F For all n > 1, A091225(a(n)) = A010060(n). [Maps odious numbers to binary representations of irreducible GF(2) polynomials (A014580) and evil numbers to the corresponding representations of reducible polynomials (A091242), in some order. A246162 has the same property].

%o (Scheme, with memoization-macro definec, two alternative definitions)

%o (definec (A246164 n) (cond ((= 1 n) n) ((= 1 (A010060 n)) (A014580 (A246164 (- (A065620 n) 1)))) (else (A091242 (A246164 (- (A065620 n)))))))

%o (definec (A246164 n) (cond ((= 1 n) n) ((= 1 (A010060 n)) (A014580 (A246164 (- (A246160 n) 1)))) (else (A091242 (A246164 (A246159 n))))))

%Y Inverse: A246163.

%Y Similar or related permutations: A246206, A246202, A193231, A245702, A234025, A246162, A234612, A246204.

%Y Cf. A010060, A014580, A048724, A065620, A065621, A091242, A246159, A246160.

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 19 2014