# Calculate Home Primes

The Home Prime of an integer $$\n\$$ is the value obtained by repeatedly factoring and concatenating $$\n\$$'s prime factors (in ascending order, including repeats) until reaching a fixed point (a prime). For example, the Home Prime ($$\\text{HP}(n)\$$) of $$\10\$$ is $$\773\$$ as:

\begin{align} 10 & = 2 \times 5 \to 25 \\ 25 & = 5 \times 5 \to 55 \\ 55 & = 5 \times 11 \to 511 \\ 511 & = 7 \times 73 \to 773 \\ \end{align}

There are two equivalent ways to consider when the sequence ends:

• It ends at a prime
• It reaches a fixed point, as the prime factors of a prime $$\p\$$ is just $$\p\$$

Note that the Home Prime of some numbers is currently unknown (e.g. $$\49\$$ or $$\77\$$).

You are to take a positive integer $$\n \ge 2\$$ as input through any convenient method or format and output the Home Prime of $$\n\$$. You may assume that you don't have to handle any input that would exceed the integer limit in your language at any step, and you may assume that the input will already have a known Home Prime (so 49 won't be an input).

Make sure you program handles all inputs correctly, not just those that are only semiprimes:

\begin{align} \text{HP}(24) = 331319 :\\ 24 & = 2 \times 2 \times 2 \times 3 \to 2223 \\ 2223 & = 3 \times 3 \times 13 \times 19 \to 331319 \end{align}

This is so the shortest code in bytes wins!

## Test cases

These are the results for each $$\2 \le n \le 100\$$, excluding $$\n = 49,77,80,96\$$ which don't terminate on TIO in my example program.

  2                                  2
3                                  3
4                                211
5                                  5
6                                 23
7                                  7
8                3331113965338635107
9                                311
10                                773
11                                 11
12                                223
13                                 13
14                              13367
15                               1129
16                           31636373
17                                 17
18                                233
19                                 19
20                3318308475676071413
21                                 37
22                                211
23                                 23
24                             331319
25                                773
26                               3251
27                              13367
28                                227
29                                 29
30                                547
31                                 31
32                             241271
33                                311
34                              31397
35                               1129
36                              71129
37                                 37
38                                373
39                                313
40                      3314192745739
41                                 41
42                                379
43                                 43
44                        22815088913
45                            3411949
46                                223
47                                 47
48             6161791591356884791277
50                               3517
51                                317
52                               2213
53                                 53
54                               2333
55                                773
56                              37463
57                               1129
58                                229
59                                 59
60                              35149
61                                 61
62                              31237
63                                337
64                      1272505013723
65 1381321118321175157763339900357651
66                               2311
67                                 67
68                               3739
69                              33191
70                                257
71                                 71
72                            1119179
73                                 73
74                                379
75                                571
76                             333271
78                         3129706267
79                                 79
81                    193089459713411
82                                241
83                                 83
84                               2237
85                               3137
86          6012903280474189529884459
87         41431881512748629379008933
88                             719167
89                                 89
90                              71171
91                          236122171
92                             331319
93                                331
94                               1319
95                              36389
97                                 97
98                                277
99                              71143
100                             317047

• Related fastest-code version. Brownie points for beating or matching my 6 byte Jelly answer Apr 14 at 9:01

# 05AB1E, 3 bytes

ΔÒJ


Try it online!

Δ run until the output doesn't change:
Ò prime factors including duplicates
J join into an integer

• WOW that is so short! Apr 15 at 11:52

# JavaScript (ES6),  58  55 bytes

f=n=>n-(g=d=>q=n>1?n%d?g(d+1):d+g(d,n/=d):'')(2)?f(q):q


Try it online!

### Commented

f = n =>               // f is a recursive function taking the input n
n - (                // subtract from n the result of a call to ...
g = d =>           // ... g: a recursive function taking a divisor d
q =                //   save in q:
n > 1 ?          //     if n is greater than 1:
n % d ?        //       if d is not a divisor of n:
g(d + 1)     //         increment d until it is
:              //       else:
d +          //         append d
g(d,         //         append the result of a recursive call
n /= d) //         with n divided by d
:                //     else:
''             //       stop and force coercion to a string
)(2)                 // initial call to g with d = 2
?                    // if it's not equal to n:
f(q)               //   recursive call to f with n = q
:                    // else:
q                  //   we're done: return q


# APL(Dyalog Extended), 12 bytes SBCS

{⍎⊃,/⍕¨⍭⍵}⍣=


Try it on APLgolf!

A dfn submission which takes a single argument.

• Lol, this was mine {⍎⊃,/⍕¨3⌂pco⍵}⍣= but apparently pco only works with 32 bit integers. Apr 14 at 21:28

# Bash, 84 82 71 bytes

Saved 11 bytes thanks to caird coinheringaahing!!!

for((;$1-${2-0};)){ set - factor $1|sed 's/.*://;s/ //g'$1;};echo $1  Try it online! Returns the home prime of $$\n\$$ quickly (performs all testcase in less than 2 seconds on TIO). • @cairdcoinheringaahing Nice one - thanks! Please post suggestions as comments not edits. Apr 15 at 19:44 • It looks like changing the condition to $10-$20 also works. Additionally, you can combine the two sed substitutions into one to save some more. Sep 10 at 8:12 # Haskell, 79 bytes until((==)<*>g)g g=read.f f 1="" f n=[show p++f(div n p)|p<-[2..],mod n p<1]!!0  Try it online! The relevant function is until((==)<*>g)g, which takes as input a number n and returns its Home prime. # Husk, 6 bytes ω(rṁsp  Try it online! ω( # iterate until reaching a fixed point: p # get the prime factors ṁs # convert each to a string & concatenate r # convert the string to a value  # Vyxal, 5 4 bytes ‡ǐṅẊ  Try it Online! Jelly really do be getting rekt by stack languages though :p This isn't 4 bytes because strings and integers aren't interchangeable like 05ab1e (and by extension Ohm), but that's okay. I added better type cohesion. ## Explained ‡ǐṅẊ ‡ǐṅ # lambda x: "".join(prime_factorisation(x)) Ẋ # repeat the above on the input until it doesn't change.  # Pyth, 14 bytes W!P_Q=QsjkPQ;Q  Try it online! # Explanation Q # integer input W!P_Q # While Q is not prime =Q # Set Q to PQ # Returns prime factors of Q in increasing order. k # Empty string jk # Join them s # Convert to integer ; # End of loop Q # Print the final value of Q  # Ohm v2, 4 bytes ·ΘoJ  Try it online! # Jelly, 6 bytes ÆfV$ÐL


Try it online!

This seems oddly long, but the prime factorize built-in is two bytes, I don't think there's a way to bypass needing the $, and the loop-until-not-unique built-in is only one byte long for the accumulator version. # R + numbers, 74 73 bytes n=scan();while(F<-el(Reduce(paste0,numbers::primeFactors(n)):0)-n)n=n+F;n  Try it online! # Japt, 12 bytes @=k ¬n)j}a;U  Try it @...}a - first number to return a truthy value when passed trough: @= > ignore input and assign 1st input U : * k ¬n) * prime factors of U joined and converted to a number j > return(is prime?) ;U - print U  • 10 bytes Apr 19 at 9:09 # Japt, 11 bytes j ?U:ßUk ¬n j ? // If the input is prime U // we're done, return the input. : // Otherwise, ß // recursively rerun Uk // with the input's prime factors ¬n // joined together as a string and parsed to a number.  Try it here. # Retina 0.8.2, 37 bytes {.+$*_
+(__+?)(\1)*.1_$#2$*_
_



Try it online! Somewhat slow, so link only includes faster test cases. Explanation:

{


Repeat until the fixed point.

.+
$*_  Convert to unary. +  Repeat until all prime factors have been found. (__+?)(\1)*$


Find the lowest factor of the remainder.

$.1_$#2$*_  Prefix the decimal of the factor to the quotient, thus concatenating it to the factors found so far. _  Delete the trailing unary 1. # Python 3, 109 bytes def f(n,m=0): while n-m: l=m=n;n=0 for i in range(2,l+1): while l%i<1:n=int(f'{n}{i}');l/=i return n  Try it online! Quite verbose, returns the home prime of $$\n\$$. # Wolfram Language (Mathematica), 60 bytes #//.x_:>FromDigits[ToString/@(""<>Table@@@FactorInteger@x)]&  Try it online! -14 bytes from @att • 60 bytes – att Apr 14 at 16:45 # Pyth, 5 bytes usjkP  Try it online! Pyth could tie 05AB1E if it had a "join into integer" byte, or Ohm if it could take the primes factors of a string. Beats Jelly even without them, which surprises me. # PowerShell, 88 bytes param($n)while($n-ne$m){$m=$n;$n=[int](((factor$m)-replace"^\d+: ").split()-join"")};$n  Try it online! Well, I'm lazy, just ported @Noodle9 # PowerShell, 203 bytes param($k)$s='$f;$n/=$f';function F($n){$m=[math]::sqrt($n);$f=2;while(!($n%$f)){iex $s};$f=3;while($f-le$m-and$n-ge$m){while(!($n%$f)){iex $s};$f+=2};$n};while($k-ne$g){$g=$k;$k=[int]((F($g))-join'')};$k


Try it online!

This one has some thought put into it...

F() is a function that returns the prime factors of a number, it works like

1. Handle 2 as a prime
2. Then brute force odd numbers
3. Stop when the factor to test exceeds the square root of the number, or the remaining quotient is less than the square root of the number
• the command factor is contained in linux only. your "lazy port" does not work with windows powershell Apr 15 at 4:22

# Scratch, 320 bytes

Try it online!

Numbers that use values greater than 1 sextillion break because Scratch automatically converts them to scientific notation. Alternatively, 38 blocks.

when gf clicked
delete all of[P v
repeat until<(length of[P v])=(1
repeat(length of[P v
set[N v]to(join(item(length of[P v])of[P v])(N
delete(length of[P v])of[P v
end
repeat until<(N)=(1
set[F v]to(2
repeat until<((N)/(F))=(round((N)/(F
change[F v]by(1
end
set[N v]to((N)/(F