A382564 -id:A382564 - cp's OEIS frontend

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

Jwalin Bhatt, Apr 29 2025

			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, ...

Cf. A002485, A382564, A383540.

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
]

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))])

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

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

A383283 Indices of record low-water marks of the sequence abs((cos n)^n).

Original entry on oeis.org

0, 1, 2, 5, 8, 11, 33, 55, 77, 99, 121, 143, 165, 187, 190, 209, 212, 234, 256, 278, 300, 322, 344, 366, 633, 655, 677, 699, 721, 1032, 1054, 1076, 1387, 1409, 1431, 1764, 2119, 2474, 2829, 3184, 3539, 3894, 4249, 4604, 4959, 5314, 5669, 6024, 6379, 6734, 7089, 7444, 7799, 8154, 8509, 8864, 9219, 9574, 9929, 10284

Offset: 0

Jwalin Bhatt, Apr 28 2025

nonn

			The first few values of abs((cos n)^n) are:
abs(cos(0)^0) = 1
abs(cos(1)^1) = 0.540302305868139
abs(cos(2)^2) = 0.173178189568194
abs(cos(3)^3) = 0.970276937921503
abs(cos(4)^4) = 0.182542548055270
abs(cos(5)^5) = 0.001836568887601
abs(cos(6)^6) = 0.783591241730686
abs(cos(7)^7) = 0.138422055397017
abs(cos(8)^8) = 0.000000200865224
abs(cos(9)^9) = 0.432737211396127
and the record low points are at n = 0, 1, 2, 5, 8, ...

C.f. A382564, A383229.

Module[{x, y, runningMin = 1.1, positions = {}},
  x = Range[0,10^6];y = Abs[Cos[x]^x];
  Do[If[y[[i]] < runningMin,runningMin = y[[i]];AppendTo[positions, i-1];],{i, Length[y]}];
  positions
]

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.

Original entry on oeis.org

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

Jwalin Bhatt, May 01 2025

nonn

			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, ...

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

Cf. A002485, A382564.

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
]

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

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A383283 Indices of record low-water marks of the sequence abs((cos n)^n).

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Author

Keywords

Examples

Crossrefs

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Mathematica

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.

Views

Author

Keywords

Examples

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Crossrefs

Programs

Mathematica