A305417 Permutation of natural numbers: a(0) = 1, a(2n) = A305421(a(n)), a(2n+1) = 2*a(n).

1, 2, 3, 4, 7, 6, 5, 8, 11, 14, 9, 12, 21, 10, 15, 16, 13, 22, 29, 28, 49, 18, 27, 24, 69, 42, 63, 20, 107, 30, 17, 32, 19, 26, 23, 44, 35, 58, 39, 56, 127, 98, 83, 36, 151, 54, 45, 48, 81, 138, 207, 84, 475, 126, 65, 40, 743, 214, 189, 60, 273, 34, 51, 64, 25, 38, 53, 52, 121, 46, 57, 88, 173, 70, 101, 116, 233, 78, 105, 112, 199, 254, 129

Offset: 0

Author

Antti Karttunen, Jun 10 2018

Keywords

• nonn
• tabf

Comments

This is GF(2)[X] analog of A005940, but note the indexing: here the domain starts from 0, although the range excludes zero.

This sequence can be represented as a binary tree. Each child to the left is obtained by applying A305421 to the parent, and each child to the right is obtained by doubling the parent:

1

|

...................2...................

3 4

7......../ \........6 5......../ \........8

/ \ / \ / \ / \

/ \ / \ / \ / \

/ \ / \ / \ / \

11 14 9 12 21 10 15 16

13 22 29 28 49 18 27 24 69 42 63 20 107 30 17 32

Sequence A305427 is obtained by scanning the same tree level by level from right to left.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16383
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A305418 (inverse), A305427 (mirror image).

Cf. A014580 (left edge from 2 onward), A305421.

Cf. also A005940, A052330, A091202.

Programs

PARI

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A305417(n) = if(0==n,(1+n),if(!(n%2),A305421(A305417(n/2)),2*(A305417((n-1)/2))));

Formula

a(0) = 1, a(2n) = A305421(a(n)), a(2n+1) = 2*a(n).

a(n) = A305427(A054429(n)).

For all n >= 1, a(A000079(n-1)) = A014580(n).