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

%I A383541 #27 May 09 2025 02:38:33
%S A383541 1,6,19,22,710,1146408,10838702,80143857,245850922,411557987,
%T A383541 1068966896
%N A383541 Positive numbers k such that (cos k)^k sets a new record.
%F A383541 Conjecture: a(n) = A002485(n+7) for n >= 9. - _Jakub Buczak_, May 05 2025
%e A383541 The first few values of (cos k)^k, k >= 1, are:
%e A383541   cos(1)^1 =  0.540302305868139
%e A383541   cos(2)^2 =  0.173178189568194
%e A383541   cos(3)^3 = -0.97027693792150
%e A383541   cos(4)^4 =  0.182542548055270
%e A383541   cos(5)^5 =  0.001836568887601
%e A383541   cos(6)^6 =  0.783591241730686
%e A383541   cos(7)^7 =  0.138422055397017
%e A383541   cos(8)^8 =  0.000000200865224
%e A383541   cos(9)^9 = -0.43273721139612
%e A383541 and the record high points are at k = 1, 6, 19, ...
%t A383541 Module[{x, y, runningMax = 0, positions = {}},
%t A383541   x = Range[1, 10^6]; y = Cos[x]^x;
%t A383541   Do[If[y[[i]] > runningMax, runningMax = y[[i]]; AppendTo[positions, i]; ], {i, Length[y]}];
%t A383541   positions
%t A383541 ]
%o A383541 (Python)
%o A383541 import numpy as np
%o A383541 x = np.arange(1, 1+10**8)
%o A383541 y = np.cos(x) ** x
%o A383541 A383541 = sorted([1+int(np.where(y==m)[0][0]) for m in set(np.maximum.accumulate(y))])
%Y A383541 Cf. A002485, A382564, A383540.
%K A383541 nonn,more
%O A383541 1,2
%A A383541 _Jwalin Bhatt_, Apr 29 2025
%E A383541 a(9)-a(11) from _Jakub Buczak_, May 05 2025