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A227455
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Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p-1 is not in the sequence.
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5
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1, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 23, 24, 25, 27, 29, 30, 33, 34, 35, 36, 39, 40, 42, 45, 46, 48, 50, 51, 53, 54, 55, 57, 58, 60, 63, 65, 66, 68, 69, 70, 72, 75, 78, 80, 81, 83, 84, 85, 87, 89, 90, 92, 93, 95, 96, 99, 100, 102, 105, 106, 108, 110
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OFFSET
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1,2
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COMMENTS
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Consider a two-player game in which players take turns and a player given the position k = p_1^s_1 * ... * p_j^s_j must choose one of the j possible moves p_1 - 1, ..., p_j - 1, and the player's chosen move becomes the position given to the other player. The first player whose only possible move is 1 loses. Terms in this sequence are the winning positions for the player whose turn it is.
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LINKS
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EXAMPLE
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Numbers of the form 2^k are not in the sequence because their unique prime divisor is p = 2 and p-1 = 1 is in the sequence.
Numbers of the form 3^k are in the sequence because 3-1 = 2 is not in the sequence.
Numbers of the form 5^k are in the sequence because 5-1 = 4 = 2^2, and 2 is not in the sequence.
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MATHEMATICA
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fa=FactorInteger; win[1] = True; win[n_] := win[n] = ! Union@Table[win[fa[n][[i, 1]] - 1], {i, 1, Length@fa@n}] == {True}; Select[Range[300], win]
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PROG
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(Haskell)
a227455 n = a227455_list !! (n-1)
a227455_list = 1 : f [2..] [1] where
f (v:vs) ws = if any (`notElem` ws) $ map (subtract 1) $ a027748_row v
then v : f vs (v : ws) else f vs ws
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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