A327138 Numbers k such that cos(2k) < cos(2k+2).

2, 5, 8, 11, 12, 14, 15, 17, 18, 20, 21, 24, 27, 30, 33, 34, 36, 37, 39, 40, 42, 43, 46, 49, 52, 55, 56, 58, 59, 61, 62, 64, 65, 68, 71, 74, 77, 78, 80, 81, 83, 84, 86, 87, 90, 93, 96, 99, 100, 102, 103, 105, 106, 108, 109, 112, 115, 118, 121, 122, 124, 125

Offset

1,1

Comments

The sequences A327138, A327139, A327140 partition the positive integers.

Conjecture: 2.07 < n*Pi - a(n) < 3.08 for n >= 1.

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Example

(cos 2, cos 4, ...) = (-0.4, -0.6, 0.9, -0.1, -0.8, ...) approximately, so that the differences, in sign, are - + - - + - - + - - + +, with "+" in places 2,5,8,11,12,... (A327138), "- +" starting in places 1,4,7,10,13,... (A327139), and "- - +" starting in places 3,6,9,22,25,... (A327140).

Mathematica

z = 500; f[x_] := f[x] = Cos[2 x]; t = Range[1, z];
Select[t, f[#] < f[# + 1] &]    (* A327138 *)
Select[t, f[#] > f[# + 1] < f[# + 2] &]  (* A327139 *)
Select[t, f[#] > f[# + 1] > f[# + 2] < f[# + 3] &]   (* A327140 *)
Position[Partition[Cos[2Range[200]], 2, 1], _?(#[[1]]<#[[2]]&), 1, Heads->False]//Flatten (* Harvey P. Dale, Oct 21 2024 *)

Crossrefs

Cf. A026309, A246303, A026317, A327139.

Sequence in context: A188802 A031141 A143263 * A102624 A070328 A124934

Adjacent sequences: A327135 A327136 A327137 * A327139 A327140 A327141

Keyword

nonn,easy

Author

Clark Kimberling, Aug 23 2019

Status

approved