A249117 - OEIS

%I #10 Aug 12 2019 02:24:00

%S 1,3,4,6,7,9,10,12,14,15,17,18,20,21,23,24,26,28,29,31,32,34,35,37,39,

%T 40,42,43,45,46,48,49,51,53,54,56,57,59,60,62,64,65,67,68,70,71,73,74,

%U 76,78,79,81,82,84,85,87,88,90,92,93,95,96,98,99,101,103

%N Position of n^6 in the ordered union of {h^6, h >= 1} and {32*k^6, k >= 1}.

%C Let S = {h^6, h >= 1} and T = {32*k^6, k >= 1}.  Then S and T are disjoint.  The position of n^6 in the ordered union of S and T is A249117(n), and the position of 32*n^6 is A249118(n).  Equivalently, the latter two give the positions of n*2^(2/3) and n*2^(3/2), respectively, when all the numbers h*2^(2/3) and k*2^(3/2) are jointly ranked.

%H Robert Israel, <a href="/A249117/b249117.txt">Table of n, a(n) for n = 1..10000</a>

%e {h^6, h >= 1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};

%e {32*k^6, k >= 1} = {32, 2048, 23328, 131072, 500000, ...};

%e so the ordered union is {1, 32, 64, 729, 2048, 4096, 15625, ...}, and a(2) = 3

%e because 2^6 is in position 3 of the ordered union.

%p Res:= NULL: count:= 0:

%p a:= 1: b:= 1:

%p for pos from 1 while count < 100 do

%p   if a^6 < 32*b^6 then

%p     Res:= Res, pos;

%p     count:= count+1;

%p     a:= a+1

%p   else

%p     b:= b+1

%p   fi

%p od:

%p Res; # _Robert Israel_, Aug 11 2019

%t z = 200; s = Table[h^6, {h, 1, z}]; t = Table[32*k^6, {k, 1, z}];

%t v = Union[s, t]  (* A249116 *)

%t Flatten[Table[Flatten[Position[v, s[[n]]]], {n, 1, 100}]]  (* A249117 *)

%t Flatten[Table[Flatten[Position[v, t[[n]]]], {n, 1, 100}]]  (* A249118 *)

%Y Cf. A249116, A249118.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 21 2014