The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A160679 Square root of n under Nim (or Conway) multiplication 1
 0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 13, 12, 9, 8, 10, 11, 30, 31, 29, 28, 25, 24, 26, 27, 16, 17, 19, 18, 23, 22, 20, 21, 57, 56, 58, 59, 62, 63, 61, 60, 55, 54, 52, 53, 48, 49, 51, 50, 39, 38, 36, 37, 32, 33, 35, 34, 41, 40, 42, 43, 46, 47, 45, 44, 124, 125, 127, 126, 123, 122, 120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Because Conway's field On2 (endowed with Nim-multiplication and [bitwise] Nim-addition) has characteristic 2, the Nim-square function (A006042) is an injective field homomorphism (i.e., the square of a sum is the sum of the squares). Thus the square function is a bijection within any finite additive subgroup of On2 (which is a fancy way to say that an integer and its Nim-square have the same bit length). Therefore the Nim square-root function is also a field homomorphism (the square-root of a Nim-sum is the Nim-sum of the square roots) which can be defined as the inverse permutation of A006042 (as such, it preserves bit-length too). LINKS Paul Tek, Table of n, a(n) for n = 0..576 FORMULA Letting NIM (= XOR) TIM and RIM denote respectively the sum, product and square root in Conway's Nim-field On2, we see that the bit-length of NIM(x,TIM(x,x)) is less than that of the positive integer x. This remark turns the following relations into an effective recursive definition of a(n) = RIM(n) which uses the fact that RIM is a field homomorphism in On2: a(0) = 0 a(n) = NIM(n, a(NIM(n, a(n, TIM(n,n)) ) Note: TIM(n,n) = A006042(n) EXAMPLE a(2) = 3 because TIM(3,3) = 2 More generally, a(x)=y because A006042(y)=x. CROSSREFS A006042 (Nim-squares). A051917 (Nim-reciprocals). Sequence in context: A340447 A154448 A099896 * A335536 A233276 A304083 Adjacent sequences:  A160676 A160677 A160678 * A160680 A160681 A160682 KEYWORD easy,nonn AUTHOR Gerard P. Michon, Jun 25 2009 STATUS approved

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

Last modified December 9 08:41 EST 2021. Contains 349627 sequences. (Running on oeis4.)