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A340780 Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.

%I A340780 #23 Feb 06 2021 12:37:34
%S A340780 1,12,18,20,28,44,45,50,52,63,68,75,76,92,98,99,116,117,120,124,147,
%T A340780 148,153,164,168,171,172,175,188,207,212,216,236,242,244,245,261,264,
%U A340780 268,270,275,279,280,284,292,312,316,325,332,333,338,356,363,369,378,387,388
%N A340780 Losing positions n (P-positions) in the following game: two players take turns dividing the current value of n by either a prime power > 1 or by A007947(n) to obtain the new value of n. The winner is the player whose division results in 1.
%C A340780 The game is equivalent to the game of Nim with the additional allowed move consisting of removing one object from each pile.
%t A340780 Clear[moves,los]; A003557[n_]:= {Module[{aux = FactorInteger[n], L=Length[FactorInteger[n]]},Product[aux[[i,1]]^(aux[[i, 2]]-1),{i, L}]]};
%t A340780 moves[n_] :=moves[n] = Module[{aux = FactorInteger[n], L=Length[ FactorInteger [n]]}, Union[Flatten[Table[n/aux[[i,1]]^j, {i,1,L},{j,1,aux[[i,2]]}],1], A003557[n]]]; los[1]=True; los[m_] := los[m] = If[PrimeQ[m], False, Union@Flatten@Table[los[moves[m][[i]]], {i,1,Length[moves[m]]}] == {False}]; Select[Range[400], los]
%Y A340780 Cf. A003557, A227691, A227763, A227764, A171947, A005240, A081691, A099352, A171945, A171949, A285304, A120442, A137295, A275432.
%K A340780 nonn
%O A340780 1,2
%A A340780 _José María Grau Ribas_, Jan 21 2021