Search: a058524 -id:a058524
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A059529
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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.
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+10
10
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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
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OFFSET
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0,3
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COMMENTS
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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)
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LINKS
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FORMULA
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a(0)=1; for 0<n, a(n) = A058377(n)+2^(n-1).
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EXAMPLE
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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
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MATHEMATICA
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Table[Length[Select[Subsets[Range[n]], Plus@@Complement[Range[n], #]>=Plus@@#&]], {n, 0, 10}] (* Gus Wiseman, Jul 04 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A060448
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Each c(i) is "multiply" (*) or "divide" (/); d(1) = 1 < d(2) < ... < d(m) = n are the divisors of n; a(n) is number of choices for c(1), ..., c(m-1) so that d(1) c(1) d(2) c(2) d(3), .., c(m-1) d(m) is an integer.
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+10
4
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1, 1, 1, 2, 1, 5, 1, 5, 2, 5, 1, 13, 1, 5, 5, 9, 1, 13, 1, 13, 5, 5, 1, 62, 2, 5, 5, 13, 1, 59, 1, 16, 5, 5, 5, 90, 1, 5, 5, 62, 1, 59, 1, 13, 13, 5, 1, 192, 2, 13, 5, 13, 1, 62, 5, 62, 5, 5, 1, 817, 1, 5, 13, 32, 5, 59, 1, 13, 5, 59, 1, 885, 1, 5, 13, 13, 5, 59, 1, 192, 9, 5, 1, 817, 5, 5
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OFFSET
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1,4
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COMMENTS
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a(n) = number of partitions of the set of divisors of n into two subsets U and V such that min(U) < min(V) and product(V) divides product(U). [Reinhard Zumkeller, Apr 05 2012]
It would appear that a(n) depends only on n's prime signature. - Charlie Neder, Oct 02 2018
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LINKS
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FORMULA
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EXAMPLE
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For n = 6 there are 5 possibilities: 1*2*3*6=36, 1/2*3*6=9, 1*2/3*6=4, 1/2/3*6=1, 1*2*3/6=1 For n = 18 there are 13 possibilities: 1*2*3*6*9*18 1/2*3*6*9*18 1*2/3*6*9*18 1*2*3/6*9*18 1*2*3*6/9*18 1*2*3*6*9/18 1/2/3*6*9*18 1/2/3*6/9*18 1/2*3*6/9*18 1*2/3/6*9*18 1*2/3*6/9*18 1*2/3*6*9/18 1*2*3/6/9*18
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PROG
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(Haskell)
import Data.List (subsequences, (\\))
a060448 n = length [us | let ds = a027750_row n,
us <- init $ tail $ subsequences ds,
let vs = ds \\ us, head us < head vs,
product us `mod` product vs == 0] + 1
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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