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Challenges

# Partial Sums of Harmonic Series

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Given an positive integer $n$, return the least positive integer $k$ such that the $k$th partial sum of the harmonic series is greater than or equal to $n$. For example, if $n = 2$, then $k = 4$, because $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{25}{12} > 2$.

# More Input/Output Examples

Input -> Output
1 -> 1
3 -> 11
4 -> 31


This is code-golf, so shortest code wins.

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# Japt, 11 10 bytes

>0©ÒßUÉ/°T


Try it

>0©ÒßUÉ/°T     :Implicit input of integer U
>0             :Greater than 0?
Ò           :Negate the bitwise NOT of (i.e., increment)
ß          :Recursive call with argument
UÉ/       :  U minus 1 divided by ...
°T     :  T (initially 0) prefix incremented
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# Python 3, 35 bytes

f=lambda x,i=1:x>0and-~f(x-1/i,i+1)


Try it online!

Subtracts each 1/i in turn from the initial value until the result is no longer positive.

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# AppleScript, 178 bytes

set n to text returned of (display dialog "" default answer "") as number
set k to 0
set h to 0
repeat
set k to k + 1
set h to h + (1 / k)
if h >= n then exit repeat
end repeat
k


It's really a shame that AppleScript doesn't support explicit stdin.

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# Raku, 27 bytes

{+([\+](1 X/1..*)...*>=$_)}  Try it online! { } # Anonymous code block [\+]( ) # Get the partial sum of 1 X/ # 1 over each of 1..* # All positive integers ...*>=$_    # Take from the list until it is bigger than the input
+(                      )   # And return the length of the list


This will start to suffer precision issues, but you can add .FatRat to work correctly (but much slower)

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# Scala, 55 bytes

(Stream from 1 map 1.0./scanLeft.0)(_+_)indexWhere _.<=


Try it online!

Written in a more sane manner:

n => Stream.from(1).map(1.0/_).scanLeft(0.0)(_+_).indexWhere(_ >= n)


Explanation:

(
Stream from 1       //Make an infinite list of integers, starting at 1
map 1.0./         //Find the reciprocal of each (divide 1.0)
scanLeft .0)(_+_) //Sum each partial sequence
indexWhere            //Find the index where
_.<=                  //n (underscore) is less than or equal to a partial sum

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# Stax, 9 bytes

Ç≈f♠É↔X+ö


Run and debug it

## Explanation(Unpacked):

wii{um|+;<
w          iterate until a falsy result is reached:
ii        push iteration number i twice
{um     map range [1..i] to their reciprocals
|+   sum that list
;< compare to the input(gets popped)
implicit output of top of stack
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# JavaScript (Node.js), 47 bytes

f=(a,b=0,c=a=>a&&1/a+c(a-1))=>c(b)<a?f(a,b+1):b


Try it online!

For large numbers, a rounding error will occur and yield incorrect result. The solution below works for any integer (no rounding errors):

# 93 bytes

f=(a,i=0n,p=a=>g(a)/c(a),g=(a,b=a)=>a&&c(b)/a+g(~-a,b),c=a=>a?a*c(~-a):1n)=>p(i)<a?f(a,-~i):i


Try it online!

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# Jelly, 8 bytes

İ€S<¬ʋ1#


Try it online!

## How it works

İ€S<¬ʋ1# - Main link. Takes n on the left
ʋ   - Group the previous 4 links into a dyad f(k, n):
€       -   Over each integer 1 to k:
İ        -     Get its reciprocal
S      -   Sum
<     -   Less than n?
¬    -   Logical not
1# - Find the first integer k ≥ n such that f(k, n) is true
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# JavaScript, 29 bytes

f=(n,x=0)=>n>0?f(n-1/++x,x):x


Try it online!

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# J, 23 bytes

>:@]^:(>1#.1%1+i.)^:_&1


Try it online!

>:@]^:(...)^:_&1   Increments from 1 while
>1#.1%1+i.         the harmonic sum up to that is less than the input

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# Goruby, 38 bytes

->n{dw{$.+=1;$..mp{1.0/-~_1}.su<n};\$.}


This was not as much fun as I thought it would be, but it was fun enough to use once.

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