A383541 Positive numbers k such that (cos k)^k sets a new record.

1, 6, 19, 22, 710, 1146408, 10838702, 80143857, 245850922, 411557987, 1068966896

Offset: 1

Author

Jwalin Bhatt, Apr 29 2025

Keywords

• nonn
• more

Examples

			The first few values of (cos k)^k, k >= 1, are:
  cos(1)^1 =  0.540302305868139
  cos(2)^2 =  0.173178189568194
  cos(3)^3 = -0.97027693792150
  cos(4)^4 =  0.182542548055270
  cos(5)^5 =  0.001836568887601
  cos(6)^6 =  0.783591241730686
  cos(7)^7 =  0.138422055397017
  cos(8)^8 =  0.000000200865224
  cos(9)^9 = -0.43273721139612
and the record high points are at k = 1, 6, 19, ...
		

Crossrefs

Cf. A002485, A382564, A383540.

Programs

Mathematica

Module[{x, y, runningMax = 0, positions = {}},
  x = Range[1, 10^6]; y = Cos[x]^x;
  Do[If[y[[i]] > runningMax, runningMax = y[[i]]; AppendTo[positions, i]; ], {i, Length[y]}];
  positions
]

Python

import numpy as np
x = np.arange(1, 1+10**8)
y = np.cos(x) ** x
A383541 = sorted([1+int(np.where(y==m)[0][0]) for m in set(np.maximum.accumulate(y))])

Formula

Conjecture: a(n) = A002485(n+7) for n >= 9. - Jakub Buczak, May 05 2025

Extensions

a(9)-a(11) from Jakub Buczak, May 05 2025