

A227455


Sequence defined recursively: 1 is in the sequence, and k > 1 is in the sequence iff for some prime divisor p of k, p1 is not in the sequence.


5



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

1,2


COMMENTS

Consider a twoplayer 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.


LINKS

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


EXAMPLE

Numbers of the form 2^k are not in the sequence because their unique prime divisor is p = 2 and p1 = 1 is in the sequence.
Numbers of the form 3^k are in the sequence because 31 = 2 is not in the sequence.
Numbers of the form 5^k are in the sequence because 51 = 4 = 2^2, and 2 is not in the sequence.


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]


PROG

(Haskell)
a227455 n = a227455_list !! (n1)
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


CROSSREFS

Cf. A227006, A227691, A027748.
Sequence in context: A187837 A325428 A239064 * A237417 A165740 A241571
Adjacent sequences: A227452 A227453 A227454 * A227456 A227457 A227458


KEYWORD

nonn


AUTHOR

José María Grau Ribas, Jul 12 2013


EXTENSIONS

Edited by Jon E. Schoenfield, Jan 23 2021


STATUS

approved



