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

%I A382564 #76 May 06 2025 10:55:21
%S A382564 1,3,22,355,5419351,411557987,1068966896,2549491779
%N A382564 Indices of records of the sequence abs((cos n)^n) starting from n = 1.
%C A382564 I conjecture that this sequence is a subsequence of the numerators of convergents to Pi (A002485).
%e A382564 The first few values of abs((cos n)^n), n >= 1, are:
%e A382564 abs(cos(1)^1) = 0.5403023058
%e A382564 abs(cos(2)^2) = 0.1731781895
%e A382564 abs(cos(3)^3) = 0.9702769379
%e A382564 abs(cos(4)^4) = 0.1825425480
%e A382564 abs(cos(5)^5) = 0.0018365688
%e A382564 and the record high points are at n = 1, 3, 22, ...
%t A382564 Module[{x, y, runningMax = 0, positions = {}},
%t A382564   x = Range[10^6]; y = Abs[Cos[x]^x];
%t A382564   Do[If[y[[i]] > runningMax, runningMax = y[[i]]; AppendTo[positions, i]; ], {i, Length[y]}];
%t A382564   positions
%t A382564 ]
%o A382564 (Python)
%o A382564 import numpy as np
%o A382564 x = np.arange(1, 1+10**8)
%o A382564 y = abs(np.cos(x) ** x)
%o A382564 A382564 = sorted([1+int(np.where(y==m)[0][0]) for m in set(np.maximum.accumulate(y))])
%o A382564 (Python)
%o A382564 from mpmath import mp
%o A382564 mp.dps = 1
%o A382564 running_max, A382564 = 0, []
%o A382564 for n in range(1, 1+10**5):
%o A382564     while ((y:=abs(mp.cos(n)**n)) == 1):
%o A382564         mp.dps += 1
%o A382564     if y > running_max:
%o A382564         running_max = y
%o A382564         A382564.append(n)
%Y A382564 C.f. A002485, A382815, A383283, A383541.
%K A382564 nonn,more
%O A382564 1,2
%A A382564 _Jwalin Bhatt_, Apr 28 2025
%E A382564 a(6)-a(8) from _Jakub Buczak_, May 04 2025