A247366 Integer parts of sqrt(2)^i * sqrt(3)^j with i,j >= 0, in natural order.

1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 16, 18, 19, 20, 22, 22, 24, 25, 27, 27, 29, 31, 32, 33, 36, 38, 39, 41, 44, 45, 46, 48, 50, 54, 55, 58, 62, 64, 66, 67, 72, 76, 78, 81, 83, 88, 90, 93, 96, 101, 108

Offset

1,4

Comments

Compare to 2^i * 3^j, the 3-smooth numbers, cf. A003586.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = floor(z*A003586(k)), k>0 and z in {1,sqrt(2),sqrt(3),sqrt(6)}.

Example

.   n | a(n) || i | j | u^i * v^j with u = sqrt(2), v = sqrt(3)
. ----+------++---+---+----------------------------------------
.   1 |    1 || 0 | 0 | u^0 * v^0 =                 1.0
.   2 |    1 || 1 | 0 | u^1 * v^0 = 1 * u =         1.414213...
.   3 |    1 || 0 | 1 | u^0 * v^1 = 1 * v =         1.732050...
.   4 |    2 || 2 | 0 | u^2 * v^0 = 2 =             2.0
.   5 |    2 || 1 | 1 | u^1 * v^1 = 1 * sqrt(6) =   2.449489...
.   6 |    2 || 3 | 0 | u^3 * v^0 = 2 * u =         2.828427...
.   7 |    3 || 0 | 2 | u^0 * v^2 = 3 =             3.0
.   8 |    3 || 2 | 1 | u^2 * v^1 = 2 * v =         3.464101...
.   9 |    4 || 4 | 0 | u^4 * v^0 = 2^2 =           4.0
.  10 |    4 || 1 | 2 | u^1 * v^2 = 3 * u =         4.242640...
.  11 |    4 || 3 | 1 | u^3 * v^1 = 2 * sqrt(6) =   4.898979...
.  12 |    5 || 0 | 3 | u^0 * v^3 = 3 * v =         5.196152...
.  13 |    5 || 5 | 0 | u^5 * v^0 = 2^2 * u =       5.656854...
.  14 |    6 || 2 | 2 | u^2 * v^2 = 2 * 3 =         6.0
.  15 |    6 || 4 | 1 | u^4 * v^1 = 2^2 * v =       6.928203...
.  16 |    7 || 1 | 3 | u^1 * v^3 = 3 * sqrt(6) =   7.348469...
.  17 |    8 || 6 | 0 | u^6 * v^0 = 2^3 =           8.0
.  18 |    8 || 3 | 2 | u^3 * v^2 = 2 * 3 * u =     8.485281...
.  19 |    9 || 0 | 4 | u^0 * v^4 = 3^2 =           9.0
.  20 |    9 || 5 | 1 | u^5 * v^1 = 2^2 * sqrt(6) = 9.797958... .

Prog

(Haskell)
import Data.Set (Set, singleton, insert, deleteFindMin)
a247366 n = a247366_list !! (n-1)
a247366_list = h $ singleton (1, 0, 0) where
   h :: Set (Double, Int, Int) -> [Integer]
   h s = (floor x) : h (insert (f i (j + 1)) $ insert (f (i + 1) j) s')
         where ((x, i, j), s') = deleteFindMin s
   f :: Int -> Int -> (Double, Int, Int)
   f u v = (2 ^^ uh * 3 ^^ vh * g ur vr, u, v) where
     g 0 0 = 1; g 0 1 = sqrt 3; g 1 0 = sqrt 2; g 1 1 = sqrt 6
     (uh, ur) = divMod u 2; (vh, vr) = divMod v 2

Crossrefs

Cf. A003586, A001951, A022838, A022840.

Sequence in context: A097508 A244225 A109964 * A285760 A025778 A294622

Adjacent sequences: A247363 A247364 A247365 * A247367 A247368 A247369

Keyword

nonn

Author

Reinhard Zumkeller, Sep 14 2014

Status

approved