The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A370717 Arithmetic derivative of elements of GF(2)[x], evaluated at x=2. 0
 0, 0, 1, 1, 0, 0, 1, 1, 4, 4, 5, 1, 4, 1, 5, 5, 0, 0, 1, 1, 0, 0, 9, 14, 4, 1, 15, 5, 4, 8, 5, 1, 16, 28, 17, 10, 16, 1, 17, 5, 20, 1, 21, 26, 4, 20, 11, 1, 16, 12, 27, 17, 4, 16, 17, 1, 20, 5, 13, 1, 20, 1, 29, 21 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS The indices of this sequence are sorted by evaluation at x=2, i.e. 0, 1, x, x+1, x^2, x^2+1, etc. Generalization of arithmetic derivative is as described by Victor Ufnarovski and Bo Ahlander. For a general UFD this requires a choice of canon primes for which p' = 1. However, GF(2)[x] has only one unit, so there is a unique arithmetic derivative over this UFD. The arithmetic derivative of a square polynomial in GF(2)[x] is 0 so square polynomials act like units: given f, g in GF(2)[x], (f*f*g)' = f*f*(g'). The arithmetic derivative of a positive degree squarefree polynomial in GF(2)[x] is always nonzero. LINKS Table of n, a(n) for n=0..63. Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003. EXAMPLE To find a(9), first convert 9 into its corresponding GF(2) polynomial x^3 + 1. Then find its arithmetic derivative, x^2. Finally, convert to an integer via evaluation at x=2, giving a(9) = 4. PROG (SageMath) P. = GF(2)[] def AD(p): F = list(p.factor()) f = [f[0] for f in F for _ in range(f[1])] return SymmetricFunctions(P).e()([len(f)-1, 0]).expand(len(f))(f) def more(l): return [x*p for p in l]+[x*p+1 for p in l] L = [x, x+1] L = L + more(L) + more(more(L)) + more(more(more(L))) + more(more(more(more(L)))) L.sort() ', '.join(map(str, ([0, 0]+[AD(p).change_ring(ZZ)(2) for p in L]))) CROSSREFS Cf. A003415, A099379, A099380. Sequence in context: A200694 A021696 A006581 * A255902 A019922 A092171 Adjacent sequences: A370714 A370715 A370716 * A370718 A370719 A370720 KEYWORD nonn AUTHOR Keith J. Bauer, Feb 27 2024 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 6 17:40 EDT 2024. Contains 374981 sequences. (Running on oeis4.)