A246162 Permutation of natural numbers: a(1) = 1, a(A000069(n)) = A014580(a(n-1)), a(A001969(n)) = A091242(a(n-1)), where A000069 and A001969 are the odious and evil numbers, and A014580 resp. A091242 are the binary coded irreducible resp. reducible polynomials over GF(2).

1, 2, 4, 3, 5, 8, 11, 7, 6, 9, 13, 14, 31, 47, 17, 25, 12, 10, 19, 15, 37, 59, 20, 21, 61, 185, 42, 319, 62, 24, 87, 137, 34, 18, 55, 16, 41, 97, 27, 22, 67, 229, 49, 415, 76, 28, 103, 29, 109, 425, 78, 1627, 222, 54, 283, 3053, 373, 79, 433, 33, 131, 647, 108, 1123, 166, 45, 203, 26, 91, 379, 71, 23

Offset

1,2

Comments

This is an instance of entanglement-permutation, where the two complementary pairs to be entangled with each other are A000069/A001969 (odious and evil numbers) and A014580/A091242 (binary codes for irreducible and reducible polynomials over GF(2)).

Because 3 is the only evil number in A014580, it implies that, apart from a(4)=3, all other odious positions contain an odious number. There are also odious numbers in some of the evil positions, precisely all the terms of A246158 in some order, together with all evil numbers larger than 3. (Permutation A246164 has the same property, except there a(7)=3.) See comments in A246161 for more details how this affects the cycle structure of these permutations.

Links

Antti Karttunen, Table of n, a(n) for n = 1..207

Index entries for sequences operating on GF(2)[X]-polynomials

Index entries for sequences that are permutations of the natural numbers

Formula

a(1) = 1, and for n > 1, if A010060(n) = 1 [i.e. n is one of the odious numbers, A000069], a(n) = A014580(a(A115384(n)-1)), otherwise, a(n) = A091242(a(A245710(n))).

As a composition of related permutations:

a(n) = A245702(A233279(n)).

a(n) = A246202(A006068(n)).

a(n) = A246164(A234612(n)).

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].

Prog

(Scheme, with memoization-macro definec)
(definec (A246162 n) (cond ((= 1 n) n) ((= 1 (A010060 n)) (A014580 (A246162 (- (A115384 n) 1)))) (else (A091242 (A246162 (A245710 n))))))

Crossrefs

Inverse: A246161.

Related permutations: A006068, A233279, A234612, A237427, A245702, A246202, A246164.

Cf. A010060, A014580, A091242, A115384, A245710.

Sequence in context: A249683 A360434 A215898 * A243573 A132193 A185910

Adjacent sequences: A246159 A246160 A246161 * A246163 A246164 A246165

Keyword

nonn

Author

Antti Karttunen, Aug 17 2014. Erroneous comment corrected Aug 20 2014

Status

approved