cp's OEIS Frontend

This sequence appears to have only '3's and '4's. Indices of '4's seem to be given by A322408, but this is true only for n < 95: then the next term in A322408 is 95, but only a(96) = 4. - M. F. Hasler, Apr 19 2019

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)