%I
%S 1,2,6,4,10,3,210,8,36,5,22,24,858,105,15,16,1870,72,9699690,40,35,11,
%T 46,12,100,429,216,840,4002,30,7130,32,33,935,21,9,160660290,4849845,
%U 143,20,20746,70,1008940218,88,360,23,2569288370,96,44100,200,2805,3432,32589158477190044730,108,55,420,1616615,2001,118,60,21594,3565
%N Under the isomorphism defined in A329329, of polynomials in GF(2)[x,y] to positive integers, a(n) is the image of the polynomial that results when x+1 is substituted for x in the polynomial with image n.
%C Under the isomorphism (defined in A329329), A059897(.,.), A329329(.,.) and A003961(.) represent polynomial addition, multiplication and multiplication by x respectively; prime(i+1) represents the polynomial x^i.
%C The equivalent sequence with y+1 substituted for y is A268385.
%C Selfinverse permutation of natural numbers. Squarefree numbers are mapped to squarefree numbers, squares are mapped to squares, and in general the sequence permutes {m : A267116(m) = k} for any k.
%C From _Peter Munn_, May 31 2020: (Start)
%C The odd numbers represent the polynomials that have x as a factor. So the odd bisection's terms represent polynomials with (x+1) as a factor. They are a permutation of A268390.
%C A193231 is an equivalent sequence with respect to GF(2)[x]. See the formula showing A019565 as the related injective homomorphism, mapping the usual encoding of GF(2) polynomials in x to their equivalent A329329defined representation.
%C (End)
%H Antti Karttunen, <a href="/A334205/b334205.txt">Table of n, a(n) for n = 1..3670</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F a(prime(i)^j) = A123098(i1)^j, a(A059897(n, k)) = A059897(a(n), a(k)).
%F a(n) = A225546(A268385(A225546(n))).
%F a(A003961(n)) = A059897(a(n), A003961(a(n))) = A329329(6, a(n)).
%F a(n^2) = a(n)^2.
%F a(n) = a(A007913(n)) * a(A008833(n)).
%F a(A019565(n)) = A019565(A193231(n)).
%F A267116(a(n)) = A267116(n).
%e Calculation for n = 5. 5 = prime(3) = prime(2+1) is the image of the polynomial x^2. Substituting x+1 for x, this becomes (x+1)^2 = x^2 + (1+1)x + 1 = x^2 + 1, as 1 + 1 = 0 in GF(2). The image of x^2 + 1 is A059897(prime(3), prime(1)) = A059897(5, 2) = 10. So a(5) = 10. (Note that A059897 gives the same result as multiplication when its operands are different terms of A050376, such as prime numbers.)
%o (PARI)
%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i])));
%o A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
%o A268385(n) = if(1==n, n, my(f=factor(n)); prod(i=1,#f~,f[i,1]^A193231(f[i,2])));
%o A334205(n) = A225546(A268385(A225546(n)));
%o (PARI)
%o \\ This program is better for larger values. A048675 and A193231 as in above:
%o A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A334205(n) = if(1==n, n, if(issquare(n), A334205(sqrtint(n))^2, A019565(A193231(A048675(core(n)))) * A334205(n/core(n)))); \\ _Antti Karttunen_, May 24 2020
%Y Cf. A123098, A267116.
%Y Equivalent GF(2)[x] sequence is A193231 (via A019565).
%Y Equivalent sequences for other substitutions: x > 0: A006519, (x > y, y > x): A225546, y > y+1: A268385, x > x^2: A319525.
%Y Cf. A268390 (ordered odd bisection).
%Y A003961, A007913, A008833, A059897, A329329 are used to express relationship between terms of this sequence.
%K nonn
%O 1,2
%A _Antti Karttunen_ and _Peter Munn_, May 04 2020
