A172515 -id:A172515 - cp's OEIS frontend

A167389 (arg(exp(-w)) + Im(w)) / (2*Pi), with w = W(n,-log(2)/2)/log(2), where W is the Lambert W function.

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

Offset: 1

Stephen Crowley, Nov 02 2009

The definition seems unnecessarily obscure. What is really going on here? - N. J. A. Sloane, Nov 13 2009
The complement is A172513 with first differences in A172515. - R. J. Mathar, Feb 27 2010
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
From Travis Scott, Oct 09 2022: (Start)
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)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Rob Corless, Poster
Rob Corless, Lambert W function, copy on web.archove.org as of 07/2011.
Stephen Crowley, A Mysterious Three Term Integer Sequence Related to a Lambert W Function Solution to a Certain Transcendental Equation [broken link?]
Eric Weisstein's World of Mathematics, Lambert W-Function.
Wikipedia, Lambert W function.

Cf. A172513 (complement).

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)

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 *)

(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) ~ n/log(2). - Vaclav Kotesovec, Jul 08 2021
a(n) = floor(Im(W(n,-log(2)/2))/(Pi*log(4))+1/2). - Travis Scott, Oct 09 2022

A172513 Complement of A167389.

Original entry on oeis.org

1, 4, 7, 11, 14, 17, 20, 24, 27, 30, 33, 37, 40, 43, 46, 50, 53, 56, 59, 63, 66, 69, 73, 76, 79, 82, 86, 89, 92, 95, 99, 102, 105, 108, 112, 115, 118, 121, 125, 128, 131, 134, 138, 141, 144, 147, 151, 154, 157, 161, 164, 167, 170, 174, 177, 180, 183, 187, 190, 193

Offset: 1

Stephen Crowley, Feb 05 2010

nonn

It appears that the sequence of first differences (A172515) consists of only 3's and 4's. - M. F. Hasler, Apr 11 2019

Cf. A167389, A172515.

sort(convert((convert([seq(n, n = 1 .. 1000)], set) minus convert([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 .. 1000)], set)), list))

A172513_upto(LIM=200)=setminus([1..LIM],A167389_upto(200)) \\ M. F. Hasler, Apr 11 2019

Keyword "hard" removed by M. F. Hasler, Apr 11 2019

cp's OEIS Frontend

A167389 (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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