# I want my book to be away from this table

Story

So I have a book that I want to separate from my table with nothing but other books. I want to know how many books do I need to achieve this with $n$ book lengths.

Here's a visualization that my friend at Wolfram drew for me:

More information about the topic in Wolfram and Wikipedia.

Challenge

Given an integer input $n$, output how many books needed for the top book to be $n$ book length away from the table horizontally.
or
Find the smallest integer value of $m$ for input $n$ in the following inequality. $$\sum_{i=1}^{m}\frac{1}{2i} \geq n$$

Edit: for fractions use at least a IEEE single-precision floating point. sorry for editing challenge after posting

(OEIS A014537)

Test cases

 1          4
2         31
3        227
5      12367
10  272400600

• youtube.com/watch?v=pBYPXsGka74 <-- related – streetster Jan 27 '18 at 16:24
• Does it have to be using this particular arrangement of books, which IIRC is not optimal? – user253751 Feb 5 '18 at 3:46

# Octave, 4140 33 bytes

1 byte saved thanks to @Dennis

@(n)find(cumsum(.5./(1:9^n))>n,1)


Try it online!

### Explanation

This uses the fact that harmonic numbers can be lower-bounded by a logarithmic function.

Also, the >= comparison can be replaced by > because harmonic numbers cannot be even integers (thanks, @Dennis!).

@(n)                                   % Anonymous function of n
1:9^n             % Range [1 2 ... 9^n]
.5./(     )            % Divide .5 by each entry
cumsum(           )           % Cumulative sum
>n         % Is each entry greater than n?
find(                     ,1)      % Index of first true entry


# Python 3, 35 bytes

f=lambda n,k=2:n>0and-~f(n-1/k,k+2)


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# Husk, 8 bytes

V≥⁰∫m\İ0


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Since Husk uses rational numbers when it can, this has no floating point issues

### Explanation

      İ0    The infinite list of positive even numbers
m\      Reciprocate each
∫        Get the cumulative sum
V           Find the index of the first element
≥⁰         that is greater than or equal to the input

• 8 bytes, but in which charset? – john16384 Jan 27 '18 at 23:03
• @john16384 Husk has it's own codepage where each symbol corresponds to a single byte. Here is the corresponding hexdump – H.PWiz Jan 27 '18 at 23:15

# JavaScript, 30 bytes

A recursive function so it'll crap out pretty early.

f=(n,x=0)=>n>0?f(n-.5/++x,x):x


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# Haskell, 38 bytes

k!n|n<=0=0|x<-n-1/(2*k)=1+(k+1)!x
(1!)


# Swift, 65 bytes

func f(n:Double){var i=0.0,s=i;while s<n{i+=1;s+=0.5/i};print(i)}


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### Ungolfed

func f(n:Double) {
var i = 0.0, s = 0.0
while s < n {
i += 1;
s += 0.5 / i
}
print(i)
}


# R, 39 bytes

function(n){while((F=F+.5/T)<n)T=T+1;T}


Try it online!

Brute Force!

# Javascript (ES6), 34 bytes

n=>eval("for(i=0;n>0;n-=.5/i)++i")


### Ungolfed

n => {
for(i = 0; n > 0; ++i)
n -= .5 / i
return i;
}


### Test Cases

f=n=>eval("for(i=0;n>0;n-=.5/i)++i")
<button onclick="console.log(f(1))">Run for n = 1</button>
<button onclick="console.log(f(2))">Run for n = 2</button>
<button onclick="console.log(f(3))">Run for n = 3</button>
<button onclick="console.log(f(5))">Run for n = 5</button>
<button onclick="console.log(f(10))">Run for n = 10</button>

• Came up with a similar solution using recursion for 30 bytes. Dunno whether to post it or not, after seeing yours. – Shaggy Jan 27 '18 at 14:14
• I may be missing something, but why do you need to wrap it in an eval statement? – caird coinheringaahing Jan 27 '18 at 22:03
• @cairdcoinherigaahing, without the eval the i variable would need to be returned at the end, at the cost of a few more bytes. – Shaggy Jan 27 '18 at 22:12

# Jelly, 8 bytes

RİSH:ð1#


This is very slow.

Try it online!

# Haskell, 7149 48 bytes

f x=length.fst.span(<x).scanl(+)0$(0.5/)<$>[1..]


@BMO saved me a whopping 22 bytes!

# Julia 0.6, 30 27 bytes

<(n,i=1)=n>0?n-.5/i<i+1:i-1


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Only works up to n = 6, because Julia has no tail call optimization.

-3 bytes thanks to Dennis.

# TI-BASIC, 27 bytes

Prompts user for input and displays output on termination. Note: ⁻¹ is the -1 (inverse) token.

Input N
1
Repeat 2N≤Σ(I⁻¹,I,1,Ans
Ans+1
End
Ans

• If you're going to save Ans in N immediately, then Input N or Prompt N is an input method that saves you one byte over Ans→N. And M can be replaced by Ans, so that 1→M becomes 1 and M+1→M becomes Ans+1. (But I am skeptical about an output in Ans that does not get displayed - see this - so maybe ending with :Ans is appropriate: then the value will be shown in place of "Done".) – Misha Lavrov Jan 31 '18 at 4:32
• Thank you! I knew Ans→N felt funny. Nice optimizations. Also took your advice on output just to be safe. Still comes out with a net -3 bytes :D – kamoroso94 Jan 31 '18 at 5:22

# 05AB1E, 11 bytes

XµN·zODI›}N


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Explanation

Xµ       }    # loop until counter is 1
N·z         # push 1/(2*N)
O        # sum the stack
DI›     # break if the sum is greater than the input
N   # push N


# Pyth, 10 bytes

f!>Qsmc1yh


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Extremely slow.

# Pyth, 10 bytes

fgscL1STyQ


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# Japt, 12 bytes

The same length as, but slightly more efficient than, the recursive option.

@T¨(Uµ½÷X}a1


Try it

## Explanation

@T¨(Uµ½÷X}a1
:Implicit input of integer U
@        }a1     :Return the first number X >=1 that returns truthy when passed through the following function
T               :Zero
¨              :Greater than or equal to
Uµ           :Decrement U by...
½÷X        :0.5 divided by X


# J, 22 bytes

-6 bytes thanks to frownyfrog

I.~0+/\@,1%2*1+[:i.9&^


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## original answer

Luis's answer in J:

1+]i.~[:<.[:+/\1%2*1+[:i.9&^


## Ungolfed

1 + ] i.~ [: <. [: +/\ 1 % 2 * 1 + [: i. 9&^


Mostly curious to see if it can be drastically improved (cough paging miles)

## Explanation

1 +      NB. 1 plus...
] i.~    NB. find the index of the arg in...
[: <.    NB. the floor of...
[: +/\   NB. the sumscan of...
1 %      NB. the reciprical of...
2 *      NB. two times...
1 +      NB. 1 plus...
[: i.    NB.  the integers up to
9&^      NB. 9 raised to the power of the arg


Try it online!

• 1+]i.~[:<. -> 1+]I.~ -> I.~0, – FrownyFrog Jan 29 '18 at 5:06
• ofc! thank you frownyfrog – Jonah Jan 29 '18 at 5:15
• And then I.~0+/\@, – FrownyFrog Jan 29 '18 at 7:43
• If you edit, you’ll beat Julia :) – FrownyFrog Jan 31 '18 at 20:29
• @FrownyFrog, done. if you have some time, i would love to see you solve this one: codegolf.stackexchange.com/questions/154345/bracket-expansion. all the solutions i can think of are too verbose to post in good conscience... – Jonah Jan 31 '18 at 22:09

# PHP, 35 bytes

while($argv[1]>$s+=.5/++$i);echo$i;


Run it using the CLI:

$php -d error_reporting=0 -r 'while($argv[1]>$s+=.5/++$i);echo\$i;' 5


# Wolfram Language (Mathematica), 40 bytes

⌈x/.NSolve[HarmonicNumber@x==2#,x]⌉&


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# Java 8, 49 bytes

n->{float r=0,s=0;for(;s<n;)s+=.5f/++r;return r;}


Explanation:

Try it online. (Times out for test cases above n=7.)

n->{             // Method with integer parameter and float return-type
float r=0,     //  Result-float, starting at 0
s=0;     //  Sum-float, starting at 0
for(;s<n;)     //  Loop as long as the sum is smaller than the input
s+=.5f/++r;  //   Increase the sum by 0.5/(r+1),
//   by first increasing r by 1 with r++
return r;}     //  Return the result-float


# tinylisp, 98 bytes

(load library
(d _(q((k # N D)(i(l N(* D # 2))(_(inc k)#(+(* N k)D)(* D k))(dec k
(q((#)(_ 1 # 0 1


The last line is an unnamed lambda function that takes the number of book-lengths and returns the number of books needed. Try it online!

### Explanation

The only numeric data type tinylisp has is integers, so we calculate the harmonic series as a fraction by keeping track of the numerator and denominator. At each step, N is the numerator, D is the denominator, and k is the sum index. We want the new partial sum to be N/D + 1/k, or (N*k + D)/(D*k). Thus, we recurse with a new numerator of N*K + D, a new denominator of D*k, and a new index of k+1.

The recursion should halt once the partial sum is greater than or equal to #, the desired number of book-lengths. At this point, we've gone one book too far, so we return k-1. The condition is 1/2 * N/D < #; multiplying out the denominator, we get N < D*#*2, which is the golfiest way to write it.

The recursive helper function _ does all these calculations; the main function is merely a one-argument wrapper that calls _ with the correct starting values for k, N, and D.