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%I #19 Nov 23 2016 11:18:15
%S 0,3,10,13,44,47,102,105,146,149,232,235,636,639,814,817,950,953,1208,
%T 1211,2994,2997,4922,4925,4996,4999,6748,6751,8026,8029,8478,8481,
%U 12092,12095,14004,14007,31934,31937,35824,35827,41568,41571,46118,46121,60056,60059,62530,62533,106986,106989
%N P-positions for the subtraction game whose allowed moves are the practical numbers (A005153).
%C According to a general theorem of Golomb (1966) on subtraction games, this sequence is infinite, and more strongly (because of known results on the density of A005153) the number of terms below any given n is at least logarithmic in n.
%H S. W. Golomb, <a href="http://dx.doi.org/10.1016/S0021-9800(66)80016-9">A mathematical investigation of games of "take-away"</a>, J. Combinatorial Theory, 1 (1966), 443-458.
%e For instance, 10 is a P-position because each of the available moves (subtracting 1, 2, 4, 6, or 8 to yield 9, 8, 6, 4, or 2) can be countered: from 8, 6, 4, or 2, it is possible to win by moving directly to 0 and from 9 it is possible to win by subtracting 6 and moving to the smaller P-position 3.
%Y Cf. A030193.
%K nonn
%O 0,2
%A _David Eppstein_, Nov 20 2016
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