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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}).

%I A019312 #111 Oct 15 2022 05:59:02
%S A019312 0,2,3,7,9,15,17,21,30,40,44,50,52,66,81,89,93,111,113,124,144,166,
%T A019312 170,182,198,224,251,279,285,301,303,319,352,386,418,442,448,486,503,
%U A019312 525,529,571,573,617,660,706,710,734,758,808,833,885,891,940
%N 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}).
%C A019312 In Germany this is called the Number Shark (Zahlenhai) sequence: see the CrypTool link.
%C A019312 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
%C A019312 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
%H A019312 Brian Chess, <a href="/A019312/b019312.txt">Table of n, a(n) for n = 1..1000</a> (Terms 1..158 by Dan Hoey; 159..227 by Timothy Loh; 228..404 by Bernhard Esslinger; 405..519 by van Nek)
%H A019312 Brian Chess, <a href="https://github.com/bvchess/taxman/wiki">taxman</a>
%H A019312 Bernhard Esslinger, <a href="http://www.cryptool.org">CrypTool</a>
%H A019312 Atli Fannar Franklín and Robert K. Moniot, <a href="/A019312/a019312.pdf">Polynomial-Time Upper Bound to the Taxman Score</a>
%H A019312 Dan Hoey, <a href="/A019312/a019312.txt">Notes on A019312</a>
%H A019312 Robert K. Moniot, <a href="http://dsm.dsm.fordham.edu/~moniot/taxman-game.pdf">The Taxman Game</a>
%H A019312 Brandee Wilson, <a href="https://web.archive.org/web/20151023160904/http://scimath.unl.edu/MIM/files/MATExamFiles/Wilson_FINAL.pdf">The Taxman Game</a>
%F A019312 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.
%o A019312 (Haskell)
%o A019312 import Data.List ((\\), intersect)
%o A019312 a019312 = t . enumFromTo 1 where
%o A019312    t xs = foldl max 0 [z + t (xs \\ ds) | z <- xs,
%o A019312              let ds = a027750_row z, not $ null $ intersect xs $ init ds]
%o A019312 -- _Reinhard Zumkeller_, Apr 05 2015
%Y A019312 Cf. A355079.
%K A019312 nonn,nice
%O A019312 1,2
%A A019312 _Dan Hoey_
%E A019312 Extended by _Timothy Loh_, Aug 12 2012