A058524 -id:A058524 - cp's OEIS frontend

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

Naohiro Nomoto, Feb 16 2001

nonn

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)

			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

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

Cf. A058524, A058377.

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

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

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

More terms from Alois P. Heinz, Jun 13 2019

A060448 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.

Original entry on oeis.org

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

Offset: 1

Naohiro Nomoto, Apr 14 2001

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

			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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Reinhard Zumkeller, Example for n = 120

Cf. A058524, A060636.

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
-- Reinhard Zumkeller, Apr 05 2012

a(A008578(n)) = 1; a(A002808(n)) > 1. [Reinhard Zumkeller, Apr 05 2012]

cp's OEIS Frontend

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.

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