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%I #28 Aug 08 2014 10:28:38
%S 1,2,3,4,7,5,11,6,8,12,25,9,13,17,10,14,47,18,19,34,15,20,31,24,16,21,
%T 62,26,55,27,137,45,22,28,42,33,37,23,29,79,59,35,87,71,36,166,41,58,
%U 30,38,54,44,61,49,32,39,99,76,319,46,91,108,89,48,200,53,97,75,40,50,203,70,67,57,78,64,43,51
%N Permutation of natural numbers: a(1) = 1, a(p_n) = A014580(a(n)), a(c_n) = A091242(a(n)), where p_n = n-th prime, c_n = n-th composite number and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomials over GF(2), respectively.
%C All the permutations A091202, A091204, A106442, A106444, A106446, A235041 share the same property that primes (A000040) are mapped bijectively to the binary representations of irreducible GF(2) polynomials (A014580) but while they determine the mapping of composites (A002808) to the corresponding binary codes of reducible polynomials (A091242) by a simple multiplicative rule, this permutation employs index-recursion also in that case.
%H Antti Karttunen, <a href="/A245703/b245703.txt">Table of n, a(n) for n = 1..10001</a>
%H <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(1) = 1, a(p_n) = A014580(a(n)) and a(c_n) = A091242(a(n)), where p_n is the n-th prime, A000040(n) and c_n is the n-th composite, A002808(n).
%F a(1) = 1, after which, if A010051(n) is 1 [i.e. n is prime], then a(n) = A014580(a(A000720(n))), otherwise a(n) = A091242(a(A065855(n))).
%F As a composition of related permutations:
%F a(n) = A245702(A135141(n)).
%F a(n) = A091204(A245821(n)).
%F Other identities. For all n >= 1, the following holds:
%F a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A091204 has the same property]
%F A091225(a(n)) = A010051(n). [Maps primes to binary representations of irreducible GF(2) polynomials, A014580, and nonprimes to union of {1} and the binary representations of corresponding reducible polynomials, A091242. The permutations A091202, A091204, A106442, A106444, A106446 and A235041 have the same property.]
%o (PARI)
%o allocatemem(123456789);
%o a014580 = vector(2^18);
%o a091242 = vector(2^22);
%o isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_
%o i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
%o A245703(n) = if(1==n, 1, if(isprime(n), a014580[A245703(primepi(n))], a091242[A245703(n-primepi(n)-1)]));
%o for(n=1, 10001, write("b245703.txt", n, " ", A245703(n)));
%o (Scheme, with memoization-macro definec)
%o (definec (A245703 n) (cond ((= 1 n) n) ((= 1 (A010051 n)) (A014580 (A245703 (A000720 n)))) (else (A091242 (A245703 (A065855 n))))))
%Y Inverse: A245704.
%Y Cf. A000040, A002808, A000720, A007097, A010051, A014580, A065855, A091225, A091230, A091242.
%Y Similar or related permutations: A091202, A091204, A106442, A106444, A106446, A235041, A135141, A245701, A245702, A245821, A245822, A244987, A245450.
%K nonn,look
%O 1,2
%A _Antti Karttunen_, Aug 02 2014
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