A091204 Factorization and index-recursion preserving isomorphism from nonnegative integers to polynomials over GF(2).

0, 1, 2, 3, 4, 7, 6, 11, 8, 5, 14, 25, 12, 19, 22, 9, 16, 47, 10, 31, 28, 29, 50, 13, 24, 21, 38, 15, 44, 61, 18, 137, 32, 43, 94, 49, 20, 55, 62, 53, 56, 97, 58, 115, 100, 27, 26, 37, 48, 69, 42, 113, 76, 73, 30, 79, 88, 33, 122, 319, 36, 41, 274, 39, 64, 121, 86, 185

Offset: 0

Author

Antti Karttunen, Jan 03 2004. Name changed Aug 16 2014

Keywords

• nonn

Comments

This "deeply multiplicative" isomorphism is one of the deep variants of A091202 which satisfies most of the same identities as the latter, but it additionally preserves also the structures where we recurse on prime's index. E.g. we have: A091230(n) = a(A007097(n)) and A061775(n) = A091238(a(n)). This is because the permutation induces itself when it is restricted to the primes: a(n) = A091227(a(A000040(n))).

On the other hand, when this permutation is restricted to the nonprime numbers (A018252), permutation A245814 is induced.

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: A091205.

Similar or related permutations: A091202, A106442, A106444, A106446, A235041, A245703, A245814, A245822.

Cf. A000040, A007097, A010051, A014580, A018252, A048720, A061775, A091225, A091230, A091238, A091242.

Programs

PARI

v014580 = vector(2^18); A014580(n) = v014580[n];
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
for(n=0, 8192, write("b091204.txt", n, " ", A091204(n)));
\\ Antti Karttunen, Aug 16 2014

Formula

a(0)=0, a(1)=1, a(p_i) = A014580(a(i)) for primes with index i and for composites a(p_i * p_j * ...) = a(p_i) X a(p_j) X ..., where X stands for carryless multiplication of GF(2)[X] polynomials (A048720).

As a composition of related permutations:

a(n) = A245703(A245822(n)).

Other identities.

For all n >= 0, the following holds:

a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A245703 has the same property]

For all n >= 1, the following holds:

A091225(a(n)) = A010051(n). [Maps primes bijectively to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242, in some order. The permutations A091202, A106442, A106444, A106446, A235041 and A245703 have the same property.]