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, 2, 6, 4, 10, 3, 210, 8, 36, 5, 22, 24, 858, 105, 15, 16, 1870, 72, 9699690, 40, 35, 11, 46, 12, 100, 429, 216, 840, 4002, 30, 7130, 32, 33, 935, 21, 9, 160660290, 4849845, 143, 20, 20746, 70, 1008940218, 88, 360, 23, 2569288370, 96, 44100, 200, 2805, 3432, 32589158477190044730, 108, 55, 420, 1616615, 2001, 118, 60, 21594, 3565

Offset: 1

Author

Antti Karttunen and Peter Munn, May 04 2020

Keywords

• nonn

Comments

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.

The equivalent sequence with y+1 substituted for y is A268385.

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.

From Peter Munn, May 31 2020: (Start)

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.

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.

(End)

Examples

			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.)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..3670
• Wikipedia, Polynomial ring
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A123098, A267116.

Equivalent GF(2)[x] sequence is A193231 (via A019565).

Equivalent sequences for other substitutions: x -> 0: A006519, (x -> y, y -> x): A225546, y -> y+1: A268385, x -> x^2: A319525.

Cf. A268390 (ordered odd bisection).

A003961, A007913, A008833, A059897, A329329 are used to express relationship between terms of this sequence.

Programs

PARI

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
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])));
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A268385(n) = if(1==n, n, my(f=factor(n)); prod(i=1,#f~,f[i,1]^A193231(f[i,2])));
A334205(n) = A225546(A268385(A225546(n)));

PARI

\\ This program is better for larger values. A048675 and A193231 as in above:
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
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

Formula

a(prime(i)^j) = A123098(i-1)^j, a(A059897(n, k)) = A059897(a(n), a(k)).

a(n) = A225546(A268385(A225546(n))).

a(A003961(n)) = A059897(a(n), A003961(a(n))) = A329329(6, a(n)).

a(n^2) = a(n)^2.

a(n) = a(A007913(n)) * a(A008833(n)).

a(A019565(n)) = A019565(A193231(n)).

A267116(a(n)) = A267116(n).