A171944 -id:A171944 - cp's OEIS frontend

A053661 For n > 1: if n is present, 2n is not.

1, 2, 3, 5, 7, 8, 9, 11, 12, 13, 15, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105

Offset: 1

Jeevan Chana Rai (Karanjit.Rai(AT)btinternet.com), Feb 16 2000

The Name line gives a property of the sequence, not a definition. The sequence can be defined simultaneously with b(n) := A171945(n) via a(n) = mex{a(i), b(i) : 0 <= i < n} (n >= 0}, b(n)=2a(n). The two sequences are complementary, hence A053661 is identical to A171944 (except for the first terms). Furthmore, A053661 is the same as A003159 except for the replacement of vile by dopey powers of 2. - Aviezri S. Fraenkel, Apr 28 2011
For n >= 2, either n = 2^k where k is odd or n = 2^k*m where m > 1 is odd and k is even (found by Kirk Bresniker and Stan Wagon). [Robert Israel, Oct 10 2010]
Subsequence of A175880; A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

Robert Israel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.

Essentially identical to A171944 and the complement of A171945.

a053661 n = a053661_list !! (n-1)
a053661_list = filter (> 0) a175880_list -- Reinhard Zumkeller, Feb 09 2011

N:= 1000: # to get all terms <= N
sort([1,seq(2^(2*i+1),i=0..(ilog2(N)-1)/2), seq(seq(2^(2*i)*(2*j+1),j=1..(N/2^(2*i)-1)/2),i=0..ilog2(N)/2)]); # Robert Israel, Jul 24 2015

Clear[T]; nn = 105; T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1 || k == 1, 1, If[k > n, T[k, n], If[n > k, T[k, Mod[n, k, 1]], -Product[T[n, i], {i, n - 1}]]]]]; DeleteCases[Table[If[T[n, n] == -1, n, ""], {n, 1, nn}], ""] (* Mats Granvik, Aug 25 2012 *)

More terms from James Sellers, Feb 22 2000

A171945 P-positions for game of misere version of Mark.

Original entry on oeis.org

1, 4, 6, 10, 14, 16, 18, 22, 24, 26, 30, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 64, 66, 70, 72, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 120, 122, 126, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 160, 162, 166, 168, 170, 174

Offset: 1

N. J. A. Sloane, Oct 29 2010

The same as A036554 except for the replacement of dopey by vile powers of 2. - Aviezri Fraenkel, Apr 28 2011

Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (1) (2012) 42-46.

Complement of A171944. Also for n>1, twice A171944. Cf. A036554.

isA171944 := proc(n)
    option remember;
    if n =0 or n =2 then
        true;
    elif isA171945(n) then
        false;
    else
        true ;
    end if;
end proc:
isA171945 := proc(n)
    option remember;
    if n = 0 or n =2 then
        false ;
    elif n = 1 then
        true;
    elif n mod 2 <> 0 then
        return false;
    elif isA171944(n/2) then
        true;
    else
        false ;
    end if;
end proc:
for n from 0 to 400 do
    if isA171945(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Mar 28 2013

lim = 210; S = {1}; A = {};
Do[d = Divisors[n]; If[Complement[d, S] != {n}, A = Append[A, n]; S = Union[S, d]], {n, 1, lim}];
A (* Jean-François Alcover, Jul 11 2019, after Peter Luschny in A171944 *)

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A053661 For n > 1: if n is present, 2n is not.

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