A383117 Indices of record high-water marks of the sequence abs((cos p)^p) where p is the numerator of the n-th convergent to Pi (A002485), starting from n = 1.

1, 2, 3, 5, 13, 17, 18, 19, 20, 22, 26, 28, 30, 32, 33, 34, 38, 39, 40, 43, 44, 46, 48, 49, 50, 52, 53, 55, 59, 62, 65, 67, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 83, 86, 88, 90, 91, 95, 97, 98, 100, 102, 103, 105, 106, 107, 109, 110, 111, 112, 114, 117, 119, 122, 123, 124, 125, 127, 129

Offset

1,2

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..586

Example

The first few values of abs((cos p)^p), are:
  | n  | p      | abs((cos p)^p)    |
  |----|--------|-------------------|
  | 1  | 1      | 0.540302305868139 |
  | 2  | 3      | 0.970276937921503 |
  | 3  | 22     | 0.999138535075440 |
  | 4  | 333    | 0.987127208495661 |
  | 5  | 355    | 0.999999838708950 |
  | 6  | 103993 | 0.999980973024431 |
  | 7  | 104348 | 0.999993669716665 |
  | 8  | 208341 | 0.999993141212817 |
  | 9  | 312689 | 0.999998684509338 |
  | 10 | 833719 | 0.999997769972478 |
and the record high points are at n = 1, 2, 3, 5, 13, ...

Mathematica

Module[{x, y, runningMax = 0, positions = {}},
  x = Join[{1}, Numerator @ Convergents[Pi, 64]]; y = Abs[Cos[x]^x];
  Do[If[y[[i]] > runningMax, runningMax = y[[i]]; AppendTo[positions, i]; ], {i, Length[y]}];
  positions
]

Prog

(Python)
from sympy import pi, continued_fraction_iterator, continued_fraction_convergents
from itertools import islice
from mpmath import iv
cf_iter = continued_fraction_convergents(continued_fraction_iterator(pi))
cf_coeffs = [1] + [f.numerator for f in islice(cf_iter, 20)]
running_max, A383117 = 0, []
for i, numer in enumerate(cf_coeffs, start=1):
    while True:
        x = abs(iv.cos(numer)**numer)
        if (comparison:=x>running_max) is None:
            iv.dps += 1
            max_coeff = cf_coeffs[A383117[-1]-1]
            running_max = abs(iv.cos(max_coeff)**max_coeff)
        else: break
    if comparison:
        A383117.append(i)
        running_max = x
print(A383117)  # Jwalin Bhatt, Jan 14 2026

Crossrefs

Cf. A002485, A382564.

Sequence in context: A268509 A260594 A228940 * A045311 A356196 A045312

Adjacent sequences: A383114 A383115 A383116 * A383118 A383119 A383120

Keyword

nonn

Author

Jwalin Bhatt, May 01 2025

Status

approved