

A019312


Taxman sequence: define T(S) by max{x+T(S \ {c : cx})}, where the max is over all x in S for which S also contains a proper divisor of x; if no such x exists, T(S)=0; set T(n)=T({1,...,n}).


1



0, 2, 3, 7, 9, 15, 17, 21, 30, 40, 44, 50, 52, 66, 81, 89, 93, 111, 113, 124, 144, 166, 170, 182, 198, 224, 251, 279, 285, 301, 303, 319, 352, 386, 418, 442, 448, 486, 503, 525, 529, 571, 573, 617, 660, 706, 710, 734, 758, 808, 833, 885, 891, 940
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OFFSET

1,2


COMMENTS

In Germany this is called the Number Shark (Zahlenhai) sequence: see the CrypTool link.
This sequence is associated with the taxman game. The open source cryptography elearning program JCrypTool (JCT) includes a tutorial and a discussion about strategies for the taxman game, and about how the numbers through 404 were found with a mixture of heuristic and optimal algorithms (van Nek). At least through 359 we could prove they are optimal.  Bernhard Esslinger, Mar 17 2015


LINKS

Dan Hoey, Timothy Loh, and Bernhard Esslinger, Table of n, a(n) for n = 1..404(Terms 1 through 158 were computed by DH; terms 159 through 227 by TL; terms through 404 by BE.)
Bernhard Esslinger, CrypTool
Dan Hoey, Notes on A019312
Robert K. Moniot, The Taxman Game
Brandee Wilson, The Taxman Game


FORMULA

When you take a number from S, you must give all its proper divisors to the tax man and there must be at least one to give; T(S) is the maximum total income.


PROG

(Haskell)
import Data.List ((\\), intersect)
a019312 = t . enumFromTo 1 where
t xs = foldl max 0 [z + t (xs \\ ds)  z < xs,
let ds = a027750_row z, not $ null $ intersect xs $ init ds]
 Reinhard Zumkeller, Apr 05 2015


CROSSREFS

Sequence in context: A215485 A096072 A014837 * A135369 A211539 A109660
Adjacent sequences: A019309 A019310 A019311 * A019313 A019314 A019315


KEYWORD

nonn,nice


AUTHOR

Dan Hoey


EXTENSIONS

Extended by Timothy Loh, Aug 12 2012


STATUS

approved



