Big numbers: Ultrafactorials

Question

Ultrafactorials

The ultrafactorials are a sequence of numbers which can be generated using the following function:

$$a(n) = n! ^ {n!}$$

The resulting values rise extremely quickly. Side note: This is entry A046882 in the OEIS. Also related are the hyperfactorials, a still quite huge, but a bit smaller sequence: A002109

Your task

Your task is to implement these numbers into your language. Your program will calculate the sum of all ultrafactorials from 0 up to inclusive n.

Input

Your program may only take one input: a number, which resembles the last \$a(n)\$ ultrafactorial to be added to the sum. The Input is assured to be positive or 0.

Output

Your output is all up to you, as long as there's the visible sum of the numbers somewhere.

Rules

• You can assume all integers, therefore integer input, and using integer counting loops to produce some results.

Test cases

Input: -1
Output: Any kind of error (because -1! is undefined), or no handling at all

Input: 0
Output: 1

Input: 1
Output: 2

Input: 2
Output: 6

Input: 3
Output: 46662

Challenge

This is code-golf, so the answer with the least length in bytes wins!

Answer 1

Mathematica, 19 bytes

Sum[n!^n!,{n,0,#}]&

Apologies for the extremely clear code ;)

Answer 2

Jelly, 6 bytes

Try it online!

‘Ḷ!*`S // Main link: Argument n (integer)
‘      // Take n, increment by 1
 Ḷ     // Range from [0..n]
  !    // Calculates factorial for each [0..n]
   *`  // Raises each [0!..n!] to the power of itself
     S // Sum the resulting array

Answer 3

05AB1E, 5 bytes

Code:

Ý!DmO

Explanation:

Ý       # Take the range [0, ..., input]
 !      # Map factorial over each element
  Dm    # Exponentiate each element to itself
    O   # Take the sum

Uses the CP-1252 encoding. Try it online!

Answer 4

R - 34 30 bytes

x=factorial(0:scan());sum(x^x)

vectorizing is nice

edit: saved 4 bytes thanks to @MickyT

Answer 5

J, 15 12 bytes

Saved 3 bytes thanks to miles!

1#.i.^~@!@,]

Explanation

1#.i.^~@!@,]  input: y
          ,]  append y to list...
   i.         [0, y)
        !@    factorial each member
     ^~@      raise each to itself
1#.           perform summation

Test cases

   f =: 1#.i.^~@!@,]
   (,. f"0) i.4
0     1
1     2
2     6
3 46662
   (,. f"0) i.6
0           1
1           2
2           6
3       46662
4  1.33374e33
5 3.17504e249

   echo"1] _90]\":f 6x
190281878633201797429473437859705759836595323046380462211756876146775419721154680216391116
383660154937824558291984804764687140715927099993629348072943551413397410741069111169123658
220861477766905534108349401724389611558474171816216027733366046875815097164882588181826712
426524007417126023680300953790645455254723360874298622143752208989152655091222094594342956
890526202094068774356589887610542642450567071133028553816930267473112879050178461179814798
008667622200592591542432361632955904924276854403585221477449385731481108378608652069211835
448555831555820393949831627809528917004144455150642180845929102272754394116905511650997561
389917179995442329297103257850695109383021080317204810134810158543814178231002423431556657
737982683316707709406053569620116083909440177269311235173671447595521339849978144493268530
780365729831790064477684808893338190825461650933123545889305523546630119181308584140916288
912561260392366609493077363059677222110731132927863243720195975705161197786520981159422881
575250362836779593393897664990291828935858671453835924398316498051705698128484688847592380
831018330553151156822298060174230201841578757499203145955456593022852288527824268115043999
037373974753999860179933517198889966353093307592136928730661270863274130109304971274296438
682725017433937245229524959283895094220677649257613358344409711070780405579776000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
000000003175042373780336892901667920556557182493442088021222004926225128381629943118937129
098831435345716937405655305190657814877412786176000000000000000000000000000000000000000000
000000000000000000000000000000000000000000001333735776850284124449081472890438

Answer 6

Perl 6, 41 38 37 bytes

{[+] ([**] [*](1..$_)xx 2 for 0..$_)}

{[+] ([**] ([*] 1..$_)xx 2 for 0..$_)}

{sum (($_**$_ with [*] 1..$_) for 0..$_)}

(Try it online.)

Explanation:

• for 0 .. $_: For each integer from 0 to the input,
• [*](1 .. $_) xx 2: calculate the factorial twice,
• [**] ...: and exponentiate the two identical factorials.
• [+] ...: Then sum all results of the loop.

Thanks to b2gills for 1 byte.

Answer 7

Cheddar, 44 37 bytes

n->(0|>n=>(i,a=(1|>i)/(*))->a**a)/(+)

Thank goats for reduce operator! I think it would've been good idea to add factorial

Try it online

Ungolfed

n -> ( 0|>n => (i, a=(1|>i) / (*)) -> a ** a) / (+)

Explanation

Note: A little outdated, will fix

n ->           // Input
( 0 |> n) =>   // Run below for each of [0, n]
    (              
      i,           // Input
      a =          // Let's keep n! in this variable `a`
         (1 |> i)  // Range from [1, n]
         / (*)     // Multiply all the items of that range
                   // `/` is reduce `(*)` is multiplication function
    ) ->
    a ** a         // A to the power of A
) / (+)        // Sum all results

Answer 8

MATL, 7 bytes

Q:Ygt^s

Try it online!

Explanation

Q         % Take input n implicitly. Add 1
 :        % Range [1 2 ... n+1]
  Yg      % Gamma function, element-wise. This gives [0! 1! ... n!]
    t^    % Rise result to itself, element-wise
      s   % Sum all values. Implicitly display

Answer 9

PHP, 49 bytes

for($f=1;$i<=$argv[1];$f*=++$i)$s+=$f**$f;echo$s;

INF for n>5 on a 64 bit system.

for large numbers, 70 bytes

while($i<=$argv[1])$s=gmp_add($s,gmp_pow($f=gmp_fac($i++),$f));echo$s;

requires PHP to be compiled with --with-gmp

Answer 10

Ruby, 64 66 bytes

->n{(o=(1..n).map{|i|a=(1..i).inject(:*);a**a}.inject(:+))?o+1:1}

Added two characters for the off-by-one bugfix (will look into shortening the inject calls later).

Answer 11

Pyth - 9 8 bytes

s^Rd.!Mh

Test Suite.

Answer 12

Haskell, 67 56 bytes

Note that this submission was made before the rules that banned builtins were removed.

p=product
a n=sum[p[x|x<-[p[1..i]],_<-[1..x]]|i<-[0..n]]

For example:

*Main> a 0
1
*Main> a 1
2
*Main> a 2
6
*Main> a 3
46662
*Main> a 4
1333735776850284124449081472890438
*Main> a 5
3175042373780336892901667920556557182493442088021222004926225128381629943118937129098831435345716937405655305190657814877412786176000000000000000000000000000000000000000000000000000000000000000000000000000000000000001333735776850284124449081472890438

Answer 13

Python 2, 73 72 bytes

import math
lambda n,f=math.factorial:sum(f(i)**f(i)for i in range(n+1))

Answer 14

PARI/GP, 19 bytes

n->sum(k=0,n,k!^k!)

Answer 15

GameMaker Language, 97 bytes

Main function (52 bytes)

for(a=0;a<=argument0;a++)b+=power(f(a),f(a))return b

Function f (45 bytes)

a=argument0 if!a return 1else return a*f(a--)

Answer 16

R, 42 35 bytes

Now that I've read the question properly, I've put the sum in.

This requires the gmp (multiple precision arithmetic) library to be available. This allows for large numbers to be handled. Otherwise anything over 5 returns INF.

This is implemented as a unnamed function to avoid the as.character that would be required to output to STDOUT through cat

function(x)sum((i=gmp::factorialZ(0:x))^i)

Example run

> f <- function(x)sum((i=gmp::factorialZ(0:x))^i)
> f(5)
Big Integer ('bigz') :
[1] 3175042373780336892901667920556557182493442088021222004926225128381629943118937129098831435345716937405655305190657814877412786176000000000000000000000000000000000000000000000000000000000000000000000000000000000000001333735776850284124449081472890438
> f(6)
Big Integer ('bigz') :
[1] 190281878633201797429473437859705759836595323046380462211756876146775419721154680216391116383660154937824558291984804764687140715927099993629348072943551413397410741069111169123658220861477766905534108349401724389611558474171816216027733366046875815097164882588181826712426524007417126023680300953790645455254723360874298622143752208989152655091222094594342956890526202094068774356589887610542642450567071133028553816930267473112879050178461179814798008667622200592591542432361632955904924276854403585221477449385731481108378608652069211835448555831555820393949831627809528917004144455150642180845929102272754394116905511650997561389917179995442329297103257850695109383021080317204810134810158543814178231002423431556657737982683316707709406053569620116083909440177269311235173671447595521339849978144493268530780365729831790064477684808893338190825461650933123545889305523546630119181308584140916288912561260392366609493077363059677222110731132927863243720195975705161197786520981159422881575250362836779593393897664990291828935858671453835924398316498051705698128484688847592380831018330553151156822298060174230201841578757499203145955456593022852288527824268115043999037373974753999860179933517198889966353093307592136928730661270863274130109304971274296438682725017433937245229524959283895094220677649257613358344409711070780405579776000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003175042373780336892901667920556557182493442088021222004926225128381629943118937129098831435345716937405655305190657814877412786176000000000000000000000000000000000000000000000000000000000000000000000000000000000000001333735776850284124449081472890438

f(9) will runs pretty well, but fills up a number of pages. A few hundred or so and 2,017,528 digits. f(10) kills the session on my machine.

Answer 17

JavaScript (ES7), 38 bytes

f=(n,a=1,i=0)=>i>n?0:a**a+f(n,a*++i,i)

Answer 18

Dyalog APL, 10 bytes

(+/!*!)0,⍳

How?

⍳ range of input

0, preceded with 0

!*! apply x! ^ x!

+/ sum

Answer 19

Golfscript, 41 bytes

{.{,{1+}%{*}*}{;1}if}:f;~.,\+{.f\f?}%{+}*

Try it online!

Answer 20

Thunno +, \$ 7 \log_{256}(96) \approx \$ 5.76 bytes

RFDz.^S

Attempt This Online!

Explanation

RFDz.^S  # implicit input
         # + flag adds one
R        # range from 0
 F       # factorial of each
   z.    # for each number:
  D  ^   #  raise it to the power of itself
      S  # sum the resulting list
         # implicit output

Answer 21

vemf, 3 5 bytes

▲↕!┴^

Explanation

▲↕    ' range [0, α)
   ┴  ' run this function:
    ^ '   exponentation
      ' with this for both inputs:
  !   '   factorial of each element in range

e.g. 3▲↕!┴^+ – try it online

Answer 22

Mathematica, 19 bytes

Sum[a!^a!,{a,0,#}]&

Anonymous function. Takes a number as input, and returns a number as output.

Answer 23

Pyke, 11 bytes

hFSBD]1*B)s

Try it here!

hF          -  for i in range (0, input+1)
  SB        -     product(range(1, i+1)
    D]1*    -    [^]*^
        B   -   product(^)
          s - sum(^)

Fun fact: Pyke doesn't have a factorial built-in because SB is only 2 bytes!

Answer 24

Haskell, 43 bytes

b n|f<-product[1..n]=f^f
a n=sum$b<$>[0..n]

Usage example: a 3-> 46662.

b calculates a single ultrafactorial and a sums all ultrafactorials from 0 to n.

Answer 25

JavaScript (ES7), 44 bytes

g=n=>n?(f=a=>a?a*f(a-1):1)(n)**f(n)+g(n-1):1

Answer 26

Python 2, 82 Bytes

x=input()
s=0
for z in range(x):
 t=1
 for i in range(z+1):t*=i+1
 s+=t**t
print s

Answer 27

Wonder, 33 bytes

@sum(->@^ f\prod rng1#0f)rng0+1#0

Usage:

(@sum(->@^ f\prod rng1#0f)rng0+1#0)3

Explanation

rng0+1#0

Create inclusive range from 0 to input.

->@^ f\prod rng1#0f

Map over the range with a function that 1) calculates the factorial of the item, 2) stores the result to f, and 3) calculates f^f.

sum

Sum.

Answer 28

TI-Basic, 13 bytes

sum(seq(A!^A!,A,0,Ans

P.S. You can replace sum(seq( with Σ( if you have a newer operating system (no size change).

Answer 29

Ruby 2, 41 bytes

->n{(1..n).reduce(s=1){|t,i|t+(s*=i)**s}}

Answer 30

Husk, 7 bytes

→ṁo´^Πḣ

Try it online!

→          One plus
 ṁ         the sum of
     Π     x!
    ^      to the power of
   ´       itself
 ṁo        for each x in
      ḣ    the range from 1 to
           the input.

It would be slightly more correct to use ṁo´^ΠΘḣ, actually adding 0 into the range, but since 0!^0!=1^1=1, this works too.