cp's OEIS Frontend

A245703 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.

%I A245703 #32 Apr 01 2025 17:52:08
%S A245703 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 A245703 62,26,55,27,137,45,22,28,42,33,37,23,29,79,59,35,87,71,36,166,41,58,
%U A245703 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 A245703 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 A245703 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 A245703 Antti Karttunen, <a href="/A245703/b245703.txt">Table of n, a(n) for n = 1..10001</a>
%H A245703 <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%H A245703 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A245703 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 A245703 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 A245703 As a composition of related permutations:
%F A245703 a(n) = A245702(A135141(n)).
%F A245703 a(n) = A091204(A245821(n)).
%F A245703 Other identities. For all n >= 1, the following holds:
%F A245703 a(A007097(n)) = A091230(n). [Maps iterates of primes to the iterates of A014580. Permutation A091204 has the same property]
%F A245703 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 A245703 (PARI)
%o A245703 allocatemem(123456789);
%o A245703 a014580 = vector(2^18);
%o A245703 a091242 = vector(2^22);
%o A245703 isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_
%o A245703 i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; a014580[i] = n, j++; a091242[j] = n); n++)
%o A245703 A245703(n) = if(1==n, 1, if(isprime(n), a014580[A245703(primepi(n))], a091242[A245703(n-primepi(n)-1)]));
%o A245703 for(n=1, 10001, write("b245703.txt", n, " ", A245703(n)));
%o A245703 (Scheme)
%o A245703 ;; With memoization-macro definec.
%o A245703 (definec (A245703 n) (cond ((= 1 n) n) ((= 1 (A010051 n)) (A014580 (A245703 (A000720 n)))) (else (A091242 (A245703 (A065855 n))))))
%Y A245703 Inverse: A245704.
%Y A245703 Cf. A000040, A002808, A000720, A007097, A010051, A014580, A065855, A091225, A091230, A091242.
%Y A245703 Similar or related permutations: A091202, A091204, A106442, A106444, A106446, A235041, A135141, A245701, A245702, A245821, A245822, A244987, A245450.
%K A245703 nonn,look
%O A245703 1,2
%A A245703 _Antti Karttunen_, Aug 02 2014