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}).

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

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. - Bernhard Esslinger, Mar 17 2015, Sep 17 2019 and May 04 2020

In order for a player to select a number in the game, at least one of the number's maximal factors must be available to be claimed by the taxman. - Brian Chess, Sep 24 2022

Links

Brian Chess, Table of n, a(n) for n = 1..1000 (Terms 1..158 by Dan Hoey; 159..227 by Timothy Loh; 228..404 by Bernhard Esslinger; 405..519 by van Nek)

Brian Chess, taxman

Bernhard Esslinger, CrypTool

Atli Fannar Franklín and Robert K. Moniot, Polynomial-Time Upper Bound to the Taxman Score

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

Cf. A355079.

Sequence in context: A096072 A305121 A014837 * A135369 A294283 A294122

Adjacent sequences: A019309 A019310 A019311 * A019313 A019314 A019315

Keyword

nonn,nice

Author

Dan Hoey

Extensions

Extended by Timothy Loh, Aug 12 2012

Status

approved