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).

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, 120, 58, 52, 32, 96, 9, 224, 40, 62, 240, 116, 25, 104, 64, 192, 57, 18, 448, 80, 124, 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

Offset: 1

Author

Antti Karttunen, Aug 19 2014

Keywords

• nonn

Comments

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).

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.

(Compare also to the comments given at A246161).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A246202.

Similar or related permutations: A245701, A246161, A006068, A054429, A193231, A246163, A246203, A237427.

Cf. A091225, A091226, A091245.

Formula

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)).

As a composition of related permutations:

a(n) = A054429(A245701(n)).

a(n) = A006068(A246161(n)).

a(n) = A193231(A246163(n)).

a(n) = A246203(A193231(n)).

Other identities:

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].