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A334205 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. 1


%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 Self-inverse 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 A329329-defined 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(i-1)^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

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Last modified July 26 01:33 EDT 2021. Contains 346294 sequences. (Running on oeis4.)