1,2

All the permutations A091203, A091205, A106443, A106445, A106447, A235042 share the same property that the binary representations of irreducible GF(2) polynomials (A014580) are mapped bijectively to the primes (A000040) but while they determine the mapping of corresponding reducible polynomials (A091242) to the composite numbers (A002808) by a simple multiplicative rule, this permutation employs index-recursion also in that case.

Antti Karttunen, Table of n, a(n) for n = 1..10001

a(1) = 1, after which, if A091225(n) is 1 [i.e. n is in A014580], then a(n) = A000040(a(A091226(n))), otherwise a(n) = A002808(a(A091245(n))).

As a composition of related permutations:

a(n) = A227413(A245701(n)).

a(n) = A245822(A091205(n)).

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

a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].

A010051(a(n)) = A091225(n). [After a(1)=1, maps binary representations of irreducible GF(2) polynomials (= A014580) to primes and the corresponding representations of reducible polynomials to composites].

(PARI)

allocatemem(123456789);

default(primelimit, 2^22)

A091226 = vector(2^22);

A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler

isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV

j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)

A091245(n) = ((n-A091226[n])-1);

A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));

for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));

(Scheme)

;; With memoization-macro definec.

(definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))

Inverse: A245703.

Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.

Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.

Sequence in context: A109852 A083197 A235262 * A260425 A352064 A277905

Adjacent sequences: A245701 A245702 A245703 * A245705 A245706 A245707

Antti Karttunen, Aug 02 2014

approved