Home primes

This task has been clarified. Its programming examples are in need of review to ensure that they still fit the requirements of the task.

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In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions.

The traditional notation has the prefix "HP" and a postfix count of the number of iterations until the Home prime is found (if the count is greater than 0).

Prime numbers are their own Home prime;

So:

   HP2 = 2
   
   HP7 = 7

If the integer obtained by concatenating increasing prime factors is not prime, iterate until you reach a prime number; the Home prime.

   HP4(2) = HP22(1) = 211
   HP4(2) = 2 × 2 => 22; HP22(1) = 2 × 11 => 211; 211 is prime  
   
   HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773
   HP10(4) = 2 × 5 => 25; HP25(3) = 5 × 5 => 55; HP55(2) = 5 × 11 => 511; HP511(1) = 7 × 73 => 773; 773 is prime

Task

Find and show here, on this page, the Home prime iteration chains for the integers 2 through 20 inclusive.

Stretch goal

Find and show the iteration chain for 65.

Impossible goal

Show the the Home prime for HP49.

See also

OEIS:A037274 - Home primes for n >= 2 OEIS:A056938 - Concatenation chain for HP49

Factor

Works with: Factor version 0.99 2021-02-05

<lang factor>USING: formatting kernel make math math.parser math.primes math.primes.factors math.ranges present prettyprint sequences sequences.extras ;

squish ( seq -- n ) [ present ] map-concat dec> ;

next ( m -- n ) factors squish ; inline

(chain) ( n -- ) [ dup prime? ] [ dup , next ] until , ;

chain ( n -- seq ) [ (chain) ] { } make ;

chain. ( n -- )

   chain [ last ] [ ] bi unclip "HP%d = " printf
   [ 1 + "HP%d(%d) = " printf ] each-index . ;

2 20 [a,b] [ chain. ] each</lang>

Output:

HP2 = 2
HP3 = 3
HP4 = HP22(1) = HP211(2) = 211
HP5 = 5
HP6 = HP23(1) = 23
HP7 = 7
HP8 = HP222(1) = HP2337(2) = HP31941(3) = HP33371313(4) = HP311123771(5) = HP7149317941(6) = HP22931219729(7) = HP112084656339(8) = HP3347911118189(9) = HP11613496501723(10) = HP97130517917327(11) = HP531832651281459(12) = HP3331113965338635107(13) = 3331113965338635107
HP9 = HP33(1) = HP311(2) = 311
HP10 = HP25(1) = HP55(2) = HP511(3) = HP773(4) = 773
HP11 = 11
HP12 = HP223(1) = 223
HP13 = 13
HP14 = HP27(1) = HP333(2) = HP3337(3) = HP4771(4) = HP13367(5) = 13367
HP15 = HP35(1) = HP57(2) = HP319(3) = HP1129(4) = 1129
HP16 = HP2222(1) = HP211101(2) = HP3116397(3) = HP31636373(4) = 31636373
HP17 = 17
HP18 = HP233(1) = 233
HP19 = 19
HP20 = HP225(1) = HP3355(2) = HP51161(3) = HP114651(4) = HP3312739(5) = HP17194867(6) = HP194122073(7) = HP709273797(8) = HP39713717791(9) = HP113610337981(10) = HP733914786213(11) = HP3333723311815403(12) = HP131723655857429041(13) = HP772688237874641409(14) = HP3318308475676071413(15) = 3318308475676071413

Raku

Using Prime::Factor from the Raku ecosystem.

<lang perl6>use Prime::Factor;

for flat 2..20, 65 -> $m {

   my (@steps, @factors) = $m;
   @steps.push: @factors.join.Int while (@factors = prime-factors @steps[*-1]) > 1;
   my $step = +@steps;
   say (@steps[0..*-2].map( { "HP$_\({--$step})" } ).join: ' = '),
       (+@steps > 1 ??  !! "HP$m"), " = ", @steps.tail;

}</lang>

Output:

HP2 = 2
HP3 = 3
HP4(2) = HP22(1) = 211
HP5 = 5
HP6(1) = 23
HP7 = 7
HP8(13) = HP222(12) = HP2337(11) = HP31941(10) = HP33371313(9) = HP311123771(8) = HP7149317941(7) = HP22931219729(6) = HP112084656339(5) = HP3347911118189(4) = HP11613496501723(3) = HP97130517917327(2) = HP531832651281459(1) = 3331113965338635107
HP9(2) = HP33(1) = 311
HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773
HP11 = 11
HP12(1) = 223
HP13 = 13
HP14(5) = HP27(4) = HP333(3) = HP3337(2) = HP4771(1) = 13367
HP15(4) = HP35(3) = HP57(2) = HP319(1) = 1129
HP16(4) = HP2222(3) = HP211101(2) = HP3116397(1) = 31636373
HP17 = 17
HP18(1) = 233
HP19 = 19
HP20(15) = HP225(14) = HP3355(13) = HP51161(12) = HP114651(11) = HP3312739(10) = HP17194867(9) = HP194122073(8) = HP709273797(7) = HP39713717791(6) = HP113610337981(5) = HP733914786213(4) = HP3333723311815403(3) = HP131723655857429041(2) = HP772688237874641409(1) = 3318308475676071413
HP65(19) = HP513(18) = HP33319(17) = HP1113233(16) = HP11101203(15) = HP332353629(14) = HP33152324247(13) = HP3337473732109(12) = HP111801316843763(11) = HP151740406071813(10) = HP31313548335458223(9) = HP3397179373752371411(8) = HP157116011350675311441(7) = HP331333391143947279384649(6) = HP11232040692636417517893491(5) = HP711175663983039633268945697(4) = HP292951656531350398312122544283(3) = HP2283450603791282934064985326977(2) = HP333297925330304453879367290955541(1) = 1381321118321175157763339900357651

Wren

Library: Wren-math

Library: Wren-big

Library: Wren-fmt

Library: Wren-ioutil

Not an easy task for Wren which lacks a fast factorization routine for 'big' integers - for now I've just modified the one I use for 'small' integers.

Manages to reach HP20 in about 78 seconds but HP65 is out of reasonable reach using the present approach. <lang ecmascript>import "/math" for Int import "/big" for BigInt import "/fmt" for Fmt import "/ioutil" for Output

// simple wheel based prime factors routine for BigInt var primeFactors = Fn.new { |n|

   var inc = [4, 2, 4, 2, 4, 6, 2, 6]
   var factors = []
   while (n%2 == 0) {
       factors.add(2)
       n = n / 2
   }
   while (n%3 == 0) {
       factors.add(3)
       n = n / 3
   }
   while (n%5 == 0) {
       factors.add(5)
       n = n / 5
   }
   var k = BigInt.new(7)
   var i = 0
   while (k * k <= n) {
       if (n%k == 0) {
           factors.add(k)
           n = n / k
       } else {
           k = k + inc[i]
           i = (i + 1) % 8
       }
   }
   if (n > 1) factors.add(n)
   return factors

}

for (i in 2..20) {

   Fmt.write("HP$-2d = ", i)
   if (Int.isPrime(i)) {
       System.print(i)
       continue
   }
   var n = 1
   var j = BigInt.new(i)
   while (true) {
       var k = primeFactors.call(j).reduce("") { |acc, f| acc + f.toString }
       j = BigInt.new(k)
       Output.fwrite("HP%(k)(%(n)) = ")
       if (j.isProbablePrime(1)) {
           System.print(k)
           break
       } else {
           n = n + 1
       }
   }

}</lang>

Output:

HP2  = 2
HP3  = 3
HP4  = HP22(1) = HP211(2) = 211
HP5  = 5
HP6  = HP23(1) = 23
HP7  = 7
HP8  = HP222(1) = HP2337(2) = HP31941(3) = HP33371313(4) = HP311123771(5) = HP7149317941(6) = HP22931219729(7) = HP112084656339(8) = HP3347911118189(9) = HP11613496501723(10) = HP97130517917327(11) = HP531832651281459(12) = HP3331113965338635107(13) = 3331113965338635107
HP9  = HP33(1) = HP311(2) = 311
HP10 = HP25(1) = HP55(2) = HP511(3) = HP773(4) = 773
HP11 = 11
HP12 = HP223(1) = 223
HP13 = 13
HP14 = HP27(1) = HP333(2) = HP3337(3) = HP4771(4) = HP13367(5) = 13367
HP15 = HP35(1) = HP57(2) = HP319(3) = HP1129(4) = 1129
HP16 = HP2222(1) = HP211101(2) = HP3116397(3) = HP31636373(4) = 31636373
HP17 = 17
HP18 = HP233(1) = 233
HP19 = 19
HP20 = HP225(1) = HP3355(2) = HP51161(3) = HP114651(4) = HP3312739(5) = HP17194867(6) = HP194122073(7) = HP709273797(8) = HP39713717791(9) = HP113610337981(10) = HP733914786213(11) = HP3333723311815403(12) = HP131723655857429041(13) = HP772688237874641409(14) = HP3318308475676071413(15) = 3318308475676071413