OFFSET
1,2
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)
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..3670
Wikipedia, Polynomial ring
FORMULA
EXAMPLE
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.)
PROG
(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
(PARI)
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen and Peter Munn, May 04 2020
STATUS
approved