OFFSET
1,2
COMMENTS
Let S = {h^6, h >= 1} and T = {2*k^6, k >= 1}. Then S and T are disjoint, and their ordered union is given by A249073. The position of n^6 in is A249123(n), and the position of 2*n^6 is A249124(n). Also, a(n) is the position of n in the joint ranking of the positive integers and the numbers k*2^(1/6), so that A249123 and A249124 are a pair of Beatty sequences.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n + floor(2^(-1/6)*n). - Robert Israel, Aug 12 2019
EXAMPLE
{h^6, h >= 1} = {1, 64, 729, 4096, 15625, 46656, 117649, ...};
{2*k^6, k >= 1} = {2, 128, 1458, 8192, 31250, 93312, ...};
so the ordered union is {1, 2, 64, 128, 729, 1458, 4096, 8192, 15625, ...}, and
a(2) = 3 because 2^6 is in position 3.
MAPLE
Res:= NULL: count:= 0:
a:= 1: b:= 1:
for pos from 1 while count < 100 do
if a^6 < 2*b^6 then
Res:= Res, pos;
count:= count+1;
a:= a+1
else
b:= b+1
fi
od:
Res; # Robert Israel, Aug 11 2019
MATHEMATICA
PROG
(PARI) a(n) = n + sqrtnint(((n^6) \ 2), 6) \\ David A. Corneth, Aug 11 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 21 2014
STATUS
approved