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Position of n^s in the joint ranking of {h^r} and {k^s}, where r = sqrt(2), s = sqrt(3), h > 1, k > 1.
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%I #4 Sep 01 2016 11:02:13

%S 2,4,7,10,12,15,18,21,24,27,30,34,37,40,43,47,50,53,57,60,64,67,71,74,

%T 78,81,85,88,92,96,99,103,107,110,114,118,122,125,129,133,137,141,144,

%U 148,152,156,160,164,168,172,176,180,184,188,192,196,200,204

%N Position of n^s in the joint ranking of {h^r} and {k^s}, where r = sqrt(2), s = sqrt(3), h > 1, k > 1.

%H Clark Kimberling, <a href="/A276209/b276209.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n + floor(n^(s/r)), with complement given by n + floor(n^(r/s)).

%e The first numbers in the joint ranking are

%e 2^r < 2^s < 3^r < 3^s < 4^r < 5^r < 4^s, so that a(n) = (2,4,7,...).

%t z = 150; r = N[Sqrt[2], 100]; s = N[Sqrt[3], 100];

%t u = Table[n + Floor[n^(s/r)], {n, 2, z}];

%t v = Table[n + Floor[n^(r/s)], {n, 2, z^(s/r)}];

%t w = Union[u, v];

%t Flatten[Table[Position[w, u[[n]]], {n, 1, z}]] (* A276209 *)

%t Flatten[Table[Position[w, v[[n]]], {n, 1, z}]] (* A276210 *)

%Y Cf. A276210 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Aug 31 2016