# Denominator of harmonic series

Earlier, we did the pseudofactorial of a number, which is the LCM of the numbers from 1 to n.

It would be useful in adding fractions together.

However, we find that the denominator of 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 is 20 instead of the pseudofactorial of 6, which is 60.

Your task is to find the denominator of 1/1 + 1/2 + ... + 1/n given positive integer n.

n result
1 1
2 2
3 6
4 12
5 60
6 20
7 140
8 280
9 2520
10 2520
11 27720
12 27720
13 360360
14 360360
15 360360
16 720720
17 12252240
18 4084080
19 77597520
20 15519504
21 5173168
22 5173168
23 118982864
24 356948592
25 8923714800
26 8923714800
27 80313433200
28 80313433200
29 2329089562800
30 2329089562800

# References

• How big of an input does it have to work for? Jun 13, 2016 at 21:12
• @BradGilbertb2gills As big as is reasonable. Jun 13, 2016 at 22:24

# M, 9 6 bytes

Thanks to FryAmTheEggman for saving 3 bytes! Code:

RİSg¹İ

M has a huge advantage here, because it works with fractions rather than floats. Explanation:

R       # Get the list [1 ... n].
İ      # Inverse each, resulting into [1/1, 1/2, 1/3, ..., 1/n].
S     # Sum it up. (86021/27720 for n=12)
g¹   # Compute the greatest common denominator with n. (1/27720 for n=12)
İ  # Calculate the inverse again. (27720 for n=12)

Uses the Jelly encoding. Try it online!.

Also, there is a 4-byte solution, which outputs a leading zero sometimes (e.g. 280 -> 0280). I'm not sure if this is allowed or not:

RİSV
• 1. The explanation of the 6-byte code isn't quite correct. computes the gratest common divisor of the fraction and n. Using g1 would probably be clearer. 2. V casts the fraction to a string and evaulates it niladically. <num>/ is (non-cumulative) reduce by a niladic operator. This is nonsense, but since there's only one number (the implicit argument 0), it simply does nothing. The next link, the denominator, is niladic, so the previous return value is printed implicitly and replaced with that nilad. Jun 13, 2016 at 19:49
• @Dennis Thanks! Fixed the explanation. Jun 13, 2016 at 20:13
• @Adnan Is there any documentation for M? May 30, 2017 at 7:12
• @Challenger5 Not that I know of. It is actually a variant of Jelly, but with arbitrary precision fractions. The Jelly documentation can be used, but be awate that a lot of features implemented in Jelly aren't implemented in M. May 30, 2017 at 7:33

# Julia, 22 bytes

An anonymous function.

n->1.//(1:n)|>sum|>den
• Same length: n->sum(inv,1//1:n).den Jun 14, 2016 at 0:29

### Ruby, 57 47 bytes

->n{(1..n).reduce{|a,i|a+1.to_r/i}.denominator}

Thanks to Kevin Lau for shortening this by ten bytes.

• Assign a variable to 1.to_r so you don't need to do string injection and conversion. Also, since Ruby's default for reduce is to use the first element as the initial, and 1/1=1, you don't need to specifically set 0 as the initial value. Jun 14, 2016 at 5:43
• In modern Ruby 1.to_r can be 1r Jan 17, 2022 at 13:58

# Mathematica, 27 bytes

An anonymous function.

Denominator@*HarmonicNumber

For example:

In[1] := (Denominator@*HarmonicNumber)[10]
Out[1] = 2520
• You could find a 26-byte solution if you dig into the chat :) Jun 13, 2016 at 17:57
• Oh! I should let Martin post that one, if he likes. This one is adorably literal, so I’ll keep it.
– Lynn
Jun 13, 2016 at 18:01
• Would you exemplify how the code is used? Jun 13, 2016 at 20:15

# Python 2, 69 67 bytes

a=b=k=r=1
exec'a=a*k+b;b*=k;k+=1;'*input()
while r*a%b:r+=1
print r

Test it on Ideone.

### How it works

Let H(n) be the sum of the multiplicative inverses of the first n positive integers. At all times, we have that a / b = 1 + H(k - 1). In fact, a, b, and k are all initialized to 1, and 1 / 1 = 1 = 1 + H(0).

We repeat the code snippet

a=a*k+b;b*=k;k+=1;

(as a string) n (input) times and execute the result. In each step, we update a, b, and k using the identity a / b + 1 / k = ak / bk + b / bk = (ak + b) / bk.

After all copies have been executed, a / b = 1 + H(n), which has the same denominator as H(n).

The fully reduced form of a / b is (a ÷ gcd(a,b)) / (b ÷ gcd(a,b)). Instead of calculating the greatest common divisor, we initialize r as 1 and keep incrementing r until ra is a multiple of b.

Clearly, this makes ra the least common multiple of a and b. Since gcd(a,b) · lcm(a,b) = ab, we have that b ÷ gcd(a,b) = lcm(a,b) ÷ a = ra ÷ a = r, making r the desired output.

Import Data.Ratio
f n=denominator$sum[1%k|k<-[1..n]] If the file is loaded into GHCI, f can be used as a function. • Presumbably you mean import lowercase? It saves a byte to use a map instead of a comprehension: sum$map(1%)[1..n]
– xnor
Jun 15, 2016 at 5:47

# Jelly, 9 bytes

Try it here.

Argument: n
! ÷R         Compute [n!÷1, n!÷2, … n!÷n].
©             (And store n! in the register.)
S        Find the sum of this list.
g®      GCD with n!.
®÷    Divide n! by this GCD.
• I believe it is possible to achieve the same bytecount without that register. Jun 13, 2016 at 22:49

# MATL, 14 13 bytes

:p:G:!/s1\&X<

Try it online!

### Explanation

For input N, the output is upper-bounded by N! (factorial of N). The code computes n/k for n = 1, ..., N! and for k = 1, ..., N. Then it sums over k, which gives the harmonic number multiplied by each n. The desired result is the index n of the first of those values that is an integer.

# 05AB1E, 8 bytes

Code:

!Ð¹L/O¿/

Explanation:

!         # Take the factorial of the input.
Ð        # Triplicate this.
¹L      # Get the list [1 ... input].
/O    # Divide and sum up.
¿   # Get the GCD of the sum and the factorial.
/  # Divide the factorial by this.

There might be some accuracy problems for n > 19 due to Python's division... Uses the CP-1252 encoding.

## Mathematica, 26 bytes

Denominator@Tr[1/Range@#]&

An unnamed function taking n as input and returning the denominator. Uses the standard trick of abusing Tr (trace) to sum the list of reciprocals.

# Husk, 6 bytes

is\ṁ\ḣ

Try it online!

Not too hard in Husk since it can do math with rational numbers. The hard part was extracting the denominator, I ended up inverting the fraction, converting it to string, and getting the first number from that string.

### Explanation

is\ṁ\ḣ
ḣ     range [1..n]
ṁ       sum the result of this function for each number:
\        inverse
\        invert
s         convert to string
i          get the first number in the string (the numerator)

## JavaScript (ES6), 88 bytes

m=>{for(d=1,i=0;i<m;d*=++i);for(n=i=0;i<m;n+=d/++i);for(g=d;g;[g,n]=[n%g,g]);return d/n}

Only works up to m=20 because of the limits of JavaScript's numeric precision.

# Pari/GP, 30 bytes

n->denominator(sum(i=1,n,1/i))

Try it online!

# Stax, 5 bytes

┬ΓΣ╞]

Run and debug it

6 bytes without packing.

# Whispers v3, 79 bytes

> Input
>> 1!
>> 2∕R
>> (1]
>> Each 3 4
>> ∑5
>> 6⊓2
>> 2∕7
>> Output 8

Try it online!

Same idea as Lynn's Jelly answer.

# Thunno, $$\ 12 \log_{256}(96) \approx \$$ 10 bytes

(Actually 9.88 bytes but that doesn't show up on the leaderboard)

Fztz!R+/SAG/

Attempt This Online!

#### Explanation

Fztz!R+/SAG/  # Implicit input
Fzt           # Get factorial and triplicate
z!R+       # Push range(1, input+1)
/S     # Divide and sum
AG/  # Get GCD and divide

# J, 20 bytes

(!%!+.[:+/!%1+i.)@x:

Based on the approach used by @Lynn's solution.

If precision is not necessary for large values of n or if we can assume n will be passed as an extended integer, suffixed by x, a shorter solution can be used for 15 bytes.

!%!+.[:+/!%1+i.

## Usage

f =: (!%!+.[:+/!%1+i.)@x:
f 30
2329089562800
(,:f"0) >: i. 15
1 2 3  4  5  6   7   8    9   10    11    12     13     14     15
1 2 6 12 60 20 140 280 2520 2520 27720 27720 360360 360360 360360

## Explanation

(!%!+.[:+/!%1+i.)@x:  Input: n
x:  Convert n into an extended integer
i.      Creates the range [0, 1, ..., n-1]
1+        Add one to each, range is now [1, 2, ..., n]
!           Get factorial of n
%          Divide n! by each value in the range [1, 2, ..., n]
[:+/            Sum those values
!                  Get n!
+.                Get gcd between n! and the sum
!                    Get n!
%                   Divide n! by the gcd and return

# Perl 6,  36  32 bytes

{([+] 1.FatRat X/1..$_).denominator} {([+] 1.FatRat X/1..$_).nude[1]}

### Explanation:

{
(
[+]        # reduce with &infix:<+>

# the following produces a Seq of Rational numbers
# 1/1, 1/2, 1/3 ... 1/n

1.FatRat # FatRat.new: 1,1
X/       # crossed using &infix:</>
1 .. $_ # Range from 1 to the input inclusive ) # resulting in a FatRat .nude # (nu)merator (de)nominator .[1] # grab the denominator } ### Test: my &hd = {([+] 1.FatRat X/1..$_).nude[1]}

say (1..10)».&hd; # (1 2 6 12 60 20 140 280 2520 2520)

say hd 100; # 2788815009188499086581352357412492142272
say chars hd 1000; # 433
say chars hd 10000; # 4345

## Hoon, 95 bytes

|=
@
=+
n=(gulf 1 +<)
=+
f=(roll n mul)
(div f d:(egcd f (roll (turn n |=(@ (div f +<))) add)))

Create list [1...n], fold over it with ++mul for the factorial, create list [n!/1, n!/2, ... n!/n] and sum it, find GCD of n! and the list, and divide the factorial by that number.

There's probably a much easier way to calculate the denominator, but I can't figure it out :/

• Oh Hoon, why your tokenizer needs so many redundant whitespaces? Jun 14, 2016 at 22:54
• All my Hoon entries look ugly because of the newlines :( Normal Hoon code uses two spaces between tokens, but one newline is shorter Jun 14, 2016 at 23:40

Python 3, 153 150 146 142 bytes

I'm sure this can golfed further. But I'm new here

f=lambda x:0**x or x*f(x-1)
z=int(input());i=f(z)
r=sum(i/y for y in range(1,z+1))
p=lambda a,b:a if b<1else not a%b+b or p(b,a%b)
print(i/p(r,i))
• Welcome to PPCG! Jun 13, 2016 at 22:30

# Axiom, 34 bytes

f(x)==denominator(sum(1/n,n=1..x))

test

(24) -> [[i,f(i)] for i in 1..30]
(24)
[[1,1], [2,2], [3,6], [4,12], [5,60], [6,20], [7,140], [8,280], [9,2520],
[10,2520], [11,27720], [12,27720], [13,360360], [14,360360], [15,360360],
[16,720720], [17,12252240], [18,4084080], [19,77597520], [20,15519504],
[21,5173168], [22,5173168], [23,118982864], [24,356948592],
[25,8923714800], [26,8923714800], [27,80313433200], [28,80313433200],
[29,2329089562800], [30,2329089562800]]
Type: List List Expression Integer

# PHP, 81 Bytes

for($p=1;$z++<$argn;$n=$n*$z+$p/$z)$p*=$z;for($t=1+$n;$p%--$t||$n%$t;);echo$p/$t;

Try it online!

# Japt, 12 bytes

l
/yNÎõ@/XÃx

Try it

l\n/yNÎõ@/XÃx     :Implicit input of integer U
l                 :Factorial
\n               :Reassign to U
/              :Divide by
y             :GCD with
N            :  Array of all inputs
Î           :  First element (i.e., original input)
õ          :  Range [1,NÎ]
@         :  Map each X
/X       :    Divide U by X
Ã      :  End map