A227455 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.

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

Offset: 1

Author

José María Grau Ribas, Jul 12 2013

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A227006, A227691, A027748.

Programs

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
-- Reinhard Zumkeller, Dec 08 2014

Mathematica

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]

Extensions

Edited by Jon E. Schoenfield, Jan 23 2021