This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A245704 #29 Apr 01 2025 17:51:55
%S A245704 1,2,3,4,6,8,5,9,12,15,7,10,13,16,21,25,14,18,19,22,26,33,38,24,11,28,
%T A245704 30,34,39,49,23,55,36,20,42,45,37,50,56,69,47,35,77,52,32,60,17,64,54,
%U A245704 70,78,94,66,51,29,105,74,48,41,84,53,27,88,76,95,106,73,125,91,72,44,140,97,100,68,58,115,75,40
%N A245704 Permutation of natural numbers: a(1) = 1, a(A014580(n)) = A000040(a(n)), a(A091242(n)) = A002808(a(n)), where A000040(n) = n-th prime, A002808(n) = n-th composite number, and A014580(n) and A091242(n) are binary codes for n-th irreducible and n-th reducible polynomial over GF(2), respectively.
%C A245704 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.
%H A245704 Antti Karttunen, <a href="/A245704/b245704.txt">Table of n, a(n) for n = 1..10001</a>
%H A245704 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A245704 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))).
%F A245704 As a composition of related permutations:
%F A245704 a(n) = A227413(A245701(n)).
%F A245704 a(n) = A245822(A091205(n)).
%F A245704 Other identities. For all n >= 1, the following holds:
%F A245704 a(A091230(n)) = A007097(n). [Maps iterates of A014580 to the iterates of primes. Permutation A091205 has the same property].
%F A245704 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].
%o A245704 (PARI)
%o A245704 allocatemem(123456789);
%o A245704 default(primelimit, 2^22)
%o A245704 A091226 = vector(2^22);
%o A245704 A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from _M. F. Hasler_
%o A245704 isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from _Charles R Greathouse IV_
%o A245704 j=0; n=2; while((n < 2^22), if(isA014580(n), A091226[n] = A091226[n-1]+1, A091226[n] = A091226[n-1]); n++)
%o A245704 A091245(n) = ((n-A091226[n])-1);
%o A245704 A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226[n])), A002808(A245704(A091245(n)))));
%o A245704 for(n=1, 10001, write("b245704.txt", n, " ", A245704(n)));
%o A245704 (Scheme)
%o A245704 ;; With memoization-macro definec.
%o A245704 (definec (A245704 n) (cond ((= 1 n) n) ((= 1 (A091225 n)) (A000040 (A245704 (A091226 n)))) (else (A002808 (A245704 (A091245 n))))))
%Y A245704 Inverse: A245703.
%Y A245704 Cf. A000040, A002808, A007097, A010051, A014580, A091225, A091226, A091230, A091242, A091245.
%Y A245704 Similar or related permutations: A245701, A245822, A227413, A091203, A091205, A106443, A106445, A106447, A235042, A244987, A245450.
%K A245704 nonn
%O A245704 1,2
%A A245704 _Antti Karttunen_, Aug 02 2014