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%I #45 Nov 22 2022 23:09:54
%S 2,3,5,6,8,9,10,12,13,15,16,18,19,21,22,23,25,26,28,29,31,32,34,35,36,
%T 38,39,41,42,44,45,47,48,49,51,52,54,55,57,58,60,61,62,64,65,67,68,70,
%U 71,72,74,75,77,78,80,81,83,84,85,87,88,90,91,93,94,96,97,98,100,101
%N (arg(exp(-w)) + Im(w)) / (2*Pi), with w = W(n,-log(2)/2)/log(2), where W is the Lambert W function.
%C The definition seems unnecessarily obscure. What is really going on here? - _N. J. A. Sloane_, Nov 13 2009
%C The complement is A172513 with first differences in A172515. - _R. J. Mathar_, Feb 27 2010
%C The original definition was: "(argument(exp(-(log(2)+W(n, -log(sqrt(2))))/log(2)))*log(2) + Im(W(n, -log(sqrt(2)))))/(2*Pi*log(2)) where W is the Lambert W function". The expression simplifies to that given in NAME. From the documents in LINKS, it appears that W(n,z) denotes the n-th branch of a complex LambertW function. It remains to understand the intended meaning of the distinction between arg(exp(z)) and Im(z). - _M. F. Hasler_, Apr 12 2019
%C From _Travis Scott_, Oct 09 2022: (Start)
%C One's first impression of this sequence and its complement (q.v.) is that of a Beatty duet. Indeed, a(n) never strays far from ceiling(n/log(2)), differing by 1 only at the 7, 16, 25, 34, 43, 50, 52, 59, ...-th terms.
%C By the identity arg(e^z) = Im(z)(mod 2*Pi)_(-Pi,Pi] -- where the subscripted range indicates an offset modulo rolling over at -Pi -> Pi rather than at 2*Pi -> 0 [this can be formalized as Im(z) - 2*Pi*ceiling((Im(z) - Pi)/(2*Pi))] -- we see that the argument component of our expression doesn't add any new information but rather acts on the imaginary component as part of a quotient device that reduces to floor(Im(w)/(2*Pi)+1/2) [or to round(Im(w)/(2*Pi)), with the caveat to always round up in the unlikely event that we encounter a half-integer].
%C Now consider v = W(n,-log(2)/2), taking the same product logarithm as for w but not dividing the result by log(2). Our expression then simply counts branch cuts and we get n. In very abusive but perhaps more visual language, if the sequence on v keeps track of the number of times the Im(W(n,-log(2)/2))-th power of the 2*Pi-th root of unity laps the negative real axis as we follow it counterclockwise around the unit circle, then the sequence on w keeps track of how many laps that would be on a circle of radius log(2) or by a log(4)*Pi-th root of unity.
%C It remains to guess if this tally has an intent or if it is a tally for tally's sake. (End)
%H Vincenzo Librandi, <a href="/A167389/b167389.txt">Table of n, a(n) for n = 1..1000</a>
%H Rob Corless, <a href="http://www.orcca.on.ca/LambertW/">Poster</a>
%H Rob Corless, <a href="http://kong.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/">Lambert W function</a>, <a href="http://web.archive.org/web/20110706212244/http://kong.apmaths.uwo.ca/~rcorless/frames/PAPERS/LambertW/"> copy on web.archove.org</a> as of 07/2011.
%H Stephen Crowley, <a href="http://www.vixra.org/abs/1203.0004">A Mysterious Three Term Integer Sequence Related to a Lambert W Function Solution to a Certain Transcendental Equation</a> [broken link?]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>.
%F (argument(exp(-(log(2) + W(n,-(1/2)*log(2)))/log(2)))*log(2) + Im(W(n,-(1/2)*log(2))))/ (2*Pi*log(2)).
%F a(n) ~ n/log(2). - _Vaclav Kotesovec_, Jul 08 2021
%F a(n) = floor(Im(W(n,-log(2)/2))/(Pi*log(4))+1/2). - _Travis Scott_, Oct 09 2022
%p seq(round(evalf((argument(exp(-(ln(2)+LambertW(n, -(1/2)*ln(2)))/ln(2)))*ln(2)+Im(LambertW(n,-(1/2)*ln(2))))/(2*Pi*ln(2)))), n = 1 .. 100)
%t a[n_] := (Arg[Exp[-(Log[2] + ProductLog[n, -1/2*Log[2]])/Log[2]]]* Log[2] + Im[ProductLog[n, -1/2*Log[2]]])/(2*Pi*Log[2]); Table[a[n] // Round, {n, 1, 70}] (* _Jean-François Alcover_, Jun 20 2013 *)
%t Table[Floor[Im@LambertW[n,-Log@2/2]/Log@4/Pi+1/2],{n,69}] (* _Travis Scott_, Oct 09 2022 *)
%Y Cf. A172513 (complement).
%K nonn,uned
%O 1,1
%A _Stephen Crowley_, Nov 02 2009
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