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A167389
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(arg(exp(-w)) + Im(w)) / (2*Pi), with w = W(n,-log(2)/2)/log(2), where W is the Lambert W function.
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5
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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, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 98, 100, 101
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OFFSET
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1,1
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COMMENTS
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The definition seems unnecessarily obscure. What is really going on here? - N. J. A. Sloane, Nov 13 2009
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
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.
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].
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.
It remains to guess if this tally has an intent or if it is a tally for tally's sake. (End)
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LINKS
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FORMULA
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(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)).
a(n) = floor(Im(W(n,-log(2)/2))/(Pi*log(4))+1/2). - Travis Scott, Oct 09 2022
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MAPLE
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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)
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MATHEMATICA
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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 *)
Table[Floor[Im@LambertW[n, -Log@2/2]/Log@4/Pi+1/2], {n, 69}] (* Travis Scott, Oct 09 2022 *)
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CROSSREFS
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KEYWORD
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nonn,uned
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AUTHOR
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STATUS
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approved
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