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For 1 < x, each c(i) is "multiply" (*) or "divide" (/); a(n) is number of choices for c(0),...,c(n-1) so that 1 c(0) x^1 c(1) x^2,.., c(n-1) x^n is an integer.
10

%I #7 Jul 04 2019 10:27:22

%S 1,1,2,5,9,16,32,68,135,256,512,1059,2110,4096,8192,16745,33425,65536,

%T 131072,266254,531924,1048576,2097152,4244214,8482454,16777216,

%U 33554432,67741466,135417620,268435456,536870912,1082015434,2163280087,4294967296,8589934592

%N For 1 < x, each c(i) is "multiply" (*) or "divide" (/); a(n) is number of choices for c(0),...,c(n-1) so that 1 c(0) x^1 c(1) x^2,.., c(n-1) x^n is an integer.

%C From _Gus Wiseman_, Jul 04 2019: (Start)

%C Also the number of subsets of {1..n} whose sum is less than or equal to the sum of their complement. For example, the a(0) = 1 through a(5) = 16 subsets are:

%C {} {} {} {} {} {}

%C {1} {1} {1} {1}

%C {2} {2} {2}

%C {3} {3} {3}

%C {1,2} {4} {4}

%C {1,2} {5}

%C {1,3} {1,2}

%C {1,4} {1,3}

%C {2,3} {1,4}

%C {1,5}

%C {2,3}

%C {2,4}

%C {2,5}

%C {3,4}

%C {1,2,3}

%C {1,2,4}

%C (End)

%F a(0)=1; for 0<n, a(n) = A058377(n)+2^(n-1).

%e x = 3: for n = 2 there are 2 possibilities: 1*3*9=27 and 1/3*9=3. For n = 4 there are 9 possibilities: 1*3*9*27*81 1/3*9*27*81 1*3/9*27*81 1/3/9*27*81 1*3*9/27*81 1*3*9*27/81 1/3*9/27*81 1/3*9*27/81 1*3/9/27*81

%t Table[Length[Select[Subsets[Range[n]],Plus@@Complement[Range[n],#]>=Plus@@#&]],{n,0,10}] (* _Gus Wiseman_, Jul 04 2019 *)

%Y Cf. A058524, A058377.

%Y Cf. A053632, A063865, A326173, A326174, A326175.

%K nonn

%O 0,3

%A _Naohiro Nomoto_, Feb 16 2001

%E More terms from _Alois P. Heinz_, Jun 13 2019