A382564 Indices of records of the sequence abs((cos n)^n) starting from n = 1.

1, 3, 22, 355, 5419351, 411557987, 1068966896, 2549491779

Offset: 1

Author

Jwalin Bhatt, Apr 28 2025

Keywords

• nonn
• more

Comments

I conjecture that this sequence is a subsequence of the numerators of convergents to Pi (A002485).

Examples

			The first few values of abs((cos n)^n), n >= 1, are:
abs(cos(1)^1) = 0.5403023058
abs(cos(2)^2) = 0.1731781895
abs(cos(3)^3) = 0.9702769379
abs(cos(4)^4) = 0.1825425480
abs(cos(5)^5) = 0.0018365688
and the record high points are at n = 1, 3, 22, ...
		

Crossrefs

C.f. A002485, A382815, A383283, A383541.

Programs

Mathematica

Module[{x, y, runningMax = 0, positions = {}},
  x = Range[10^6]; y = Abs[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 = abs(np.cos(x) ** x)
A382564 = sorted([1+int(np.where(y==m)[0][0]) for m in set(np.maximum.accumulate(y))])

Python

from mpmath import mp
mp.dps = 1
running_max, A382564 = 0, []
for n in range(1, 1+10**5):
    while ((y:=abs(mp.cos(n)**n)) == 1):
        mp.dps += 1
    if y > running_max:
        running_max = y
        A382564.append(n)

Extensions

a(6)-a(8) from Jakub Buczak, May 04 2025