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A019312 Taxman sequence: define T(S) by max{x+T(S \ {c : c|x})}, 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 (list; graph; refs; listen; history; text; internal format)
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 e-learning 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

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Last modified May 22 15:20 EDT 2015. Contains 257744 sequences.