proposed

approved

editing

proposed

allocated for Clark KimberlingPosition of n^r in the joint ranking of {h^r} and {k^s}, where r = sqrt(2), s = sqrt(3), h > 1, k > 1.

1, 3, 5, 6, 8, 9, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 84, 86, 87, 89, 90, 91, 93, 94, 95

1,2

Clark Kimberling, <a href="/A276210/b276210.txt">Table of n, a(n) for n = 1..10000</a>

a(n) = n + floor(n^(r/s)); the complement is given by n + floor(n^(s/r)).

The first numbers in the joint ranking are

2^r < 2^s < 3^r < 3^s < 4^r < 5^r < 4^s, so that a(n) = (1,3,5,6,...).

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

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

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

w = Union[u, v];

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

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

Cf. A276209 (complement).

allocated

nonn,easy

Clark Kimberling, Aug 31 2016

approved

editing

allocated for Clark Kimberling

allocated

approved