A059529 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.

1, 1, 2, 5, 9, 16, 32, 68, 135, 256, 512, 1059, 2110, 4096, 8192, 16745, 33425, 65536, 131072, 266254, 531924, 1048576, 2097152, 4244214, 8482454, 16777216, 33554432, 67741466, 135417620, 268435456, 536870912, 1082015434, 2163280087, 4294967296, 8589934592

Offset: 0

Author

Naohiro Nomoto, Feb 16 2001

Keywords

• nonn

Comments

From Gus Wiseman, Jul 04 2019: (Start)

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:

{} {} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{3} {3} {3}

{1,2} {4} {4}

{1,2} {5}

{1,3} {1,2}

{1,4} {1,3}

{2,3} {1,4}

{1,5}

{2,3}

{2,4}

{2,5}

{3,4}

{1,2,3}

{1,2,4}

(End)

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A058524, A058377.

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

Programs

Mathematica

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

Formula

a(0)=1; for 0A058377(n)+2^(n-1).

Extensions

More terms from Alois P. Heinz, Jun 13 2019