A275432 P-positions for the subtraction game whose allowed moves are the practical numbers (A005153).

0, 3, 10, 13, 44, 47, 102, 105, 146, 149, 232, 235, 636, 639, 814, 817, 950, 953, 1208, 1211, 2994, 2997, 4922, 4925, 4996, 4999, 6748, 6751, 8026, 8029, 8478, 8481, 12092, 12095, 14004, 14007, 31934, 31937, 35824, 35827, 41568, 41571, 46118, 46121, 60056, 60059, 62530, 62533, 106986, 106989

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..49.

S. W. Golomb, A mathematical investigation of games of "take-away", J. Combinatorial Theory, 1 (1966), 443-458.

Example

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.

Crossrefs

Cf. A030193.

Sequence in context: A285181 A042331 A082975 * A041985 A081519 A041121

Adjacent sequences: A275429 A275430 A275431 * A275433 A275434 A275435

Keyword

nonn

Author

David Eppstein, Nov 20 2016

Status

approved