Home primes

Home primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

This page uses content from Wikipedia. The original article was at Home prime. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)

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 iteration count (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 with the iteration count until you reach a prime number; The Home prime.

   HP4 = HP22(1) = 211
   HP4 = 2 × 2 => 22; HP22(1) = 2 × 11 => 211; 211 is prime  
   
   HP10 = HP25(1) = HP55(2) = HP511(3) = 773
   HP10 = 2 × 5 => 25; HP25(1) = 5 × 5 => 55; HP55(2) = 5 × 11 => 511; HP511(3) = 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%2d = " printf
   [ 1 + "HP%d(%d) = " printf ] each-index . ;

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

Output:

HP 2 = 2
HP 3 = 3
HP 4 = HP22(1) = HP211(2) = 211
HP 5 = 5
HP 6 = HP23(1) = 23
HP 7 = 7
HP 8 = 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
HP 9 = 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

<lang perl6>use Prime::Factor;

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

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

}</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
HP65 = HP513(1) = HP33319(2) = HP1113233(3) = HP11101203(4) = HP332353629(5) = HP33152324247(6) = HP3337473732109(7) = HP111801316843763(8) = HP151740406071813(9) = HP31313548335458223(10) = HP3397179373752371411(11) = HP157116011350675311441(12) = HP331333391143947279384649(13) = HP11232040692636417517893491(14) = HP711175663983039633268945697(15) = HP292951656531350398312122544283(16) = HP2283450603791282934064985326977(17) = HP333297925330304453879367290955541(18) = HP1381321118321175157763339900357651(19) = 1381321118321175157763339900357651