A091203 Factorization-preserving isomorphism from binary codes of GF(2) polynomials to integers.

0, 1, 2, 3, 4, 9, 6, 5, 8, 15, 18, 7, 12, 11, 10, 27, 16, 81, 30, 13, 36, 25, 14, 33, 24, 17, 22, 45, 20, 21, 54, 19, 32, 57, 162, 55, 60, 23, 26, 63, 72, 29, 50, 51, 28, 135, 66, 31, 48, 35, 34, 243, 44, 39, 90, 37, 40, 99, 42, 41, 108, 43, 38, 75, 64, 225, 114, 47, 324

Offset: 0

Author

Antti Karttunen, Jan 03 2004

Keywords

• nonn

Comments

E.g. we have the following identities: A000040(n) = a(A014580(n)), A091219(n) = A008683(a(n)), A091220(n) = A000005(a(n)), A091221(n) = A001221(a(n)), A091222(n) = A001222(a(n)), A091225(n) = A010051(a(n)), A091227(n) = A049084(a(n)), A091247(n) = A066247(a(n)).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• A. Karttunen, Scheme-program for computing this sequence.
• Index entries for sequences operating on GF(2)[X]-polynomials
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse: A091202.

Several variants exist: A235042, A091205, A106443, A106445, A106447.

Cf. also A000005, A000040, A001221, A001222, A004247, A008683, A010051, A014580, A048720, A049084, A066247, A091219, A091220, A091221, A091222, A091225, A091227, A091247, A234741, A234742, A245703, A245704, A278233.

Cf. also A302024, A302026, A305418, A305428 for other similar permutations.

Programs

PARI

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };
A091203(n) = if(n<=1,n,if(!(n%2),2*A091203(n/2),A003961(A091203(A305422(n))))); \\ Antti Karttunen, Jun 10 2018

Formula

a(0)=0, a(1)=1. For n's coding an irreducible polynomial ir_i, that is if n=A014580(i), we have a(n) = A000040(i) and for composite polynomials a(ir_i X ir_j X ...) = p_i * p_j * ..., where p_i = A000040(i) and X stands for carryless multiplication of GF(2)[X] polynomials (A048720) and * for the ordinary multiplication of integers (A004247).

Other identities. For all n >= 1, the following holds:

A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials, A014580, to primes and the binary representations of corresponding reducible polynomials, A091242, to composite numbers. The permutations A091205, A106443, A106445, A106447, A235042 and A245704 have the same property.]

From Antti Karttunen, Jun 10 2018: (Start)

For n <= 1, a(n) = n, for n > 1, a(n) = 2*a(n/2) if n is even, and if n is odd, then a(n) = A003961(a(A305422(n))).

a(n) = A005940(1+A305418(n)) = A163511(A305428(n)).

A046523(a(n)) = A278233(n).

(End)