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# 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 count of the number of iterations until the home prime is found (if the count is greater than 0), for instance HP4(2) === HP22(1) === 211 is the same as saying the home prime of 4 needs 2 iterations and is the same as the home prime of 22 which needs 1 iteration, and (both) resolve to 211, a prime.

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

## Factor

Works with: Factor version 0.99 2021-02-05
`USING: formatting kernel make math math.parser math.primesmath.primes.factors math.ranges present prettyprint sequencessequences.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 ; : prime. ( n -- ) dup "HP%d = %d\n" printf ; : setup ( seq -- n s r ) unclip-last swap dup length 1 [a,b] ; : multi. ( n -- ) chain setup [ "HP%d(%d) = " printf ] 2each . ; : chain. ( n -- ) dup prime? [ prime. ] [ multi. ] if ; 2 20 [a,b] [ chain. ] each`
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
```

## J

`step =: -.&' '&.":@q:hseq =: [,\$:@step`(0&\$)@.(1&p:)fmtHP =: (' is prime',~":@])`('HP',":@],'(',":@[,')'&[)@.(*@[)fmtlist =: [:;@}.[:,(<' = ')&,"0@(|[email protected]@# fmtHP each [)printHP =: 0 0&[email protected]@([email protected],(10{a.)&[)printHP"0 [ 2}.i.21exit 0`
Output:
```2 is prime
3 is prime
HP4(2) = HP22(1) = 211 is prime
5 is prime
HP6(1) = 23 is prime
7 is prime
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 is prime
HP9(2) = HP33(1) = 311 is prime
HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773 is prime
11 is prime
HP12(1) = 223 is prime
13 is prime
HP14(5) = HP27(4) = HP333(3) = HP3337(2) = HP4771(1) = 13367 is prime
HP15(4) = HP35(3) = HP57(2) = HP319(1) = 1129 is prime
HP16(4) = HP2222(3) = HP211101(2) = HP3116397(1) = 31636373 is prime
17 is prime
HP18(1) = 233 is prime
19 is prime
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 is prime```

## Julia

`using Primes function homeprimechain(n::BigInt)    isprime(n) && return [n]    concat = prod(string(i)^j for (i, j) in factor(n).pe)    return pushfirst!(homeprimechain(parse(BigInt, concat)), n)endhomeprimechain(n::Integer) = homeprimechain(BigInt(n)) function printHPiter(n, numperline = 4)    chain = homeprimechain(n)    len = length(chain)    for (i, ent) in enumerate(chain)        print(i < len ? "HP\$ent" * "(\$(len - i)) = " * (i % numperline == 0 ? "\n" : "") : "\$ent is prime.\n\n")    endend for i in [2:20; 65]   print("Home Prime chain for \$i: ")   printHPiter(i)end `
Output:
```Home Prime chain for 2: 2 is prime.

Home Prime chain for 3: 3 is prime.

Home Prime chain for 4: HP4(2) = HP22(1) = 211 is prime.

Home Prime chain for 5: 5 is prime.

Home Prime chain for 6: HP6(1) = 23 is prime.

Home Prime chain for 7: 7 is prime.

Home Prime chain for 8: 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 is prime.

Home Prime chain for 9: HP9(2) = HP33(1) = 311 is prime.

Home Prime chain for 10: HP10(4) = HP25(3) = HP55(2) = HP511(1) =
773 is prime.

Home Prime chain for 11: 11 is prime.

Home Prime chain for 12: HP12(1) = 223 is prime.

Home Prime chain for 13: 13 is prime.

Home Prime chain for 14: HP14(5) = HP27(4) = HP333(3) = HP3337(2) =
HP4771(1) = 13367 is prime.

Home Prime chain for 15: HP15(4) = HP35(3) = HP57(2) = HP319(1) =
1129 is prime.

Home Prime chain for 16: HP16(4) = HP2222(3) = HP211101(2) = HP3116397(1) =
31636373 is prime.

Home Prime chain for 17: 17 is prime.

Home Prime chain for 18: HP18(1) = 233 is prime.

Home Prime chain for 19: 19 is prime.

Home Prime chain for 20: 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 is prime.

Home Prime chain for 65: 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 is prime.
```

## Nim

Translation of: Wren
Library: bignum

This algorithm is really efficient. We get the result for HP2 to HP20 in about 6 ms and adding HP65 in 1.3 s. I think that the threshold to switch to Pollard-Rho is very important.

`import algorithm, sequtils, strformat, strutilsimport bignum let  Two = newInt(2)  Three = newInt(3)  Five = newInt(5)  proc primeFactorsWheel(n: Int): seq[Int] =  const Inc = [4, 2, 4, 2, 4, 6, 2, 6]  var n = n  while (n mod 2).isZero:    result.add Two    n = n div 2  while (n mod 3).isZero:    result.add Three    n = n div 3  while (n mod 5).isZero:    result.add Five    n = n div 5  var k = 7  var i = 0  while k * k <= n:    if (n mod k).isZero:      result.add newInt(k)      n = n div k    else:      inc k, Inc[i]      i = (i + 1) and 7  if n > 1: result.add n  func pollardRho(n : Int): Int =   func g(x, y: Int): Int = (x * x + 1) mod y   var x, y = newInt(2)  var z, d = newInt(1)  var count = 0  while true:    x = g(x, n)    y = g(g(y, n), n)    d = abs(x - y) mod n    z *= d    inc count    if count == 100:      d = gcd(z, n)      if d != 1: break      z = newInt(1)      count = 0  if d == n: return newInt(0)  result = d  proc primeFactors(n: Int): seq[Int] =  var n = n  while n > 1:    if n > 100_000_000:      let d = pollardRho(n)      if not d.isZero:        result.add primeFactorsWheel(d)        n = n div d        if n.probablyPrime(25) != 0:          result.add n          break      else:        result.add primeFactorsWheel(n)        break    else:      result.add primeFactorsWheel(n)      break  result.sort()  let list = toSeq(2..20) & 65for i in list:  if i in [2, 3, 5, 7, 11, 13, 17, 19]:    echo &"HP{i} = {i}"    continue  var n = 1  var j = newInt(i)  var h = @[j]  while true:    j = newInt(primeFactors(j).join())    h.add j    if j.probablyPrime(25) != 0:      for k in countdown(n, 1):        stdout.write &"HP{h[n-k]}({k}) = "      echo h[n]      break    else:      inc n`
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```

## Perl

Library: ntheory
`use strict;use warnings;use ntheory 'factor'; for my \$m (2..20, 65) {    my (@steps, @factors) = \$m;    push @steps, join '_', @factors while (@factors = factor \$steps[-1] =~ s/_//gr) > 1;    my \$step = \$#steps;    if (\$step >= 1) { print 'HP' . \$_ . "(\$step) = " and --\$step or last for @steps }    else            { print "HP\$m = " }    print "\$steps[-1]\n";}`
Output:
```HP2 = 2
HP3 = 3
HP4(2) = HP2_2(1) = 2_11
HP5 = 5
HP6(1) = 2_3
HP7 = 7
HP8(13) = HP2_2_2(12) = HP2_3_37(11) = HP3_19_41(10) = HP3_3_3_7_13_13(9) = HP3_11123771(8) = HP7_149_317_941(7) = HP229_31219729(6) = HP11_2084656339(5) = HP3_347_911_118189(4) = HP11_613_496501723(3) = HP97_130517_917327(2) = HP53_1832651281459(1) = 3_3_3_11_139_653_3863_5107
HP9(2) = HP3_3(1) = 3_11
HP10(4) = HP2_5(3) = HP5_5(2) = HP5_11(1) = 7_73
HP11 = 11
HP12(1) = 2_2_3
HP13 = 13
HP14(5) = HP2_7(4) = HP3_3_3(3) = HP3_3_37(2) = HP47_71(1) = 13_367
HP15(4) = HP3_5(3) = HP5_7(2) = HP3_19(1) = 11_29
HP16(4) = HP2_2_2_2(3) = HP2_11_101(2) = HP3_11_6397(1) = 3_163_6373
HP17 = 17
HP18(1) = 2_3_3
HP19 = 19
HP20(15) = HP2_2_5(14) = HP3_3_5_5(13) = HP5_11_61(12) = HP11_4651(11) = HP3_3_12739(10) = HP17_194867(9) = HP19_41_22073(8) = HP709_273797(7) = HP3_97_137_17791(6) = HP11_3610337981(5) = HP7_3391_4786213(4) = HP3_3_3_3_7_23_31_1815403(3) = HP13_17_23_655857429041(2) = HP7_7_2688237874641409(1) = 3_31_8308475676071413
HP65(19) = HP5_13(18) = HP3_3_3_19(17) = HP11_13_233(16) = HP11_101203(15) = HP3_3_23_53629(14) = HP3_3_1523_24247(13) = HP3_3_3_7_47_3732109(12) = HP11_18013_16843763(11) = HP151_740406071813(10) = HP3_13_13_54833_5458223(9) = HP3_3_97_179_373_7523_71411(8) = HP1571_1601_1350675311441(7) = HP3_3_13_33391_143947_279384649(6) = HP11_23_204069263_6417517893491(5) = HP7_11_1756639_83039633268945697(4) = HP29_29_5165653_13503983_12122544283(3) = HP228345060379_1282934064985326977(2) = HP3_3_3_2979253_3030445387_9367290955541(1) = 1381_3211183211_75157763339900357651```

## Phix

No stretch goal here this time, just the basic task. Note that mpz_prime_factors() needs a maxprime of 12207 or more to cover the 130517 needed for HP8, and in fact the 20000 used below was found by doubling a failing 10000, which...

```with javascript_semantics
include mpfr.e

procedure test(integer n)
string s = sprintf("%d",n), lastp = ""
sequence res = {s}
while true do
s = substitute(s,"_","")
sequence rr = mpz_prime_factors(s,20000)
if length(rr[\$])=1 then
lastp = rr[\$]
mpz p = mpz_init(lastp)
if not mpz_prime(p)
or length(rr)=1 then
exit
end if
rr = rr[1..\$-1]
lastp = "_"&lastp
elsif length(rr)=1
and rr=1 then
exit
end if
s = ""
for i=1 to length(rr) do
atom {prime,pow} = rr[i]
string si = sprintf("_%d",prime)
for p=1 to pow do
s &= si
end for
end for
if length(lastp) then
s &= lastp
lastp = ""
end if
res = append(res,s[2..\$])
end while
integer niter = length(res)-1
string iter = iff(niter>1?sprintf("(%d)",niter):"")
s = sprintf("HP%d%s = ",{n,iter})
if niter=0 then
printf(1,"%s%d\n",{s,n})
else
for i=2 to niter do
niter -= 1
printf(1,"%sHP%s(%d)\n",{s,res[i],niter})
if i=2 then
s = repeat(' ',length(s))
s[-2] = '='
end if
end for
printf(1,"%s%s\n",{s,res[\$]})
end if
end procedure
papply(tagset(20,2),test)
```
Output:

Using underscores to show the individual factors that were concatenated together

```HP2 = 2
HP3 = 3
HP4(2) = HP2_2(1)
= 2_11
HP5 = 5
HP6 = 2_3
HP7 = 7
HP8(13) = HP2_2_2(12)
= HP2_3_37(11)
= HP3_19_41(10)
= HP3_3_3_7_13_13(9)
= HP3_11123771(8)
= HP7_149_317_941(7)
= HP229_31219729(6)
= HP11_2084656339(5)
= HP3_347_911_118189(4)
= HP11_613_496501723(3)
= HP97_130517_917327(2)
= HP53_1832651281459(1)
= 3_3_3_11_139_653_3863_5107
HP9(2) = HP3_3(1)
= 3_11
HP10(4) = HP2_5(3)
= HP5_5(2)
= HP5_11(1)
= 7_73
HP11 = 11
HP12 = 2_2_3
HP13 = 13
HP14(5) = HP2_7(4)
= HP3_3_3(3)
= HP3_3_37(2)
= HP47_71(1)
= 13_367
HP15(4) = HP3_5(3)
= HP5_7(2)
= HP3_19(1)
= 11_29
HP16(4) = HP2_2_2_2(3)
= HP2_11_101(2)
= HP3_11_6397(1)
= 3_163_6373
HP17 = 17
HP18 = 2_3_3
HP19 = 19
HP20(15) = HP2_2_5(14)
= HP3_3_5_5(13)
= HP5_11_61(12)
= HP11_4651(11)
= HP3_3_12739(10)
= HP17_194867(9)
= HP19_41_22073(8)
= HP709_273797(7)
= HP3_97_137_17791(6)
= HP11_3610337981(5)
= HP7_3391_4786213(4)
= HP3_3_3_3_7_23_31_1815403(3)
= HP13_17_23_655857429041(2)
= HP7_7_2688237874641409(1)
= 3_31_8308475676071413
```

## Raku

Not the fastest, but not too bad either. Make an abortive attempt at HP49.

Assuming there are n steps; HP49(n - 25) is slow, HP49(n - 31) is really slow, and I gave up on HP49(n - 34) after 45 minutes.

Using Prime::Factor from the Raku ecosystem.

`use Prime::Factor; my \$start = now; (flat 2..20, 65).map: -> \$m {    my (\$now, @steps, @factors) = now, \$m;     @steps.push: @factors.join('_') while (@factors = prime-factors @steps[*-1].Int) > 1;     say (my \$step = +@steps) > 1 ?? (@steps[0..*-2].map( { "HP\$_\({--\$step})" } ).join: ' = ') !! ("HP\$m"),      " = ", @steps[*-1], "  ({(now - \$now).fmt("%0.3f")} seconds)";} say "Total elapsed time: {(now - \$start).fmt("%0.3f")} seconds\n"; say 'HP49:';my (\$now, @steps, @factors) = now, 49;my \$step = 0;while (@factors = prime-factors @steps[*-1].Int) > 1 {    @steps.push: @factors.join('_');    say "HP{@steps[\$step].Int}\(n - {\$step++}) = ", @steps[*-1], "  ({(now - \$now).fmt("%0.3f")} seconds)";    \$now = now;    last if \$step > 30;}`
Output:
```HP2 = 2  (0.000 seconds)
HP3 = 3  (0.000 seconds)
HP4(2) = HP2_2(1) = 2_11  (0.001 seconds)
HP5 = 5  (0.000 seconds)
HP6(1) = 2_3  (0.000 seconds)
HP7 = 7  (0.000 seconds)
HP8(13) = HP2_2_2(12) = HP2_3_37(11) = HP3_19_41(10) = HP3_3_3_7_13_13(9) = HP3_11123771(8) = HP7_149_317_941(7) = HP229_31219729(6) = HP11_2084656339(5) = HP3_347_911_118189(4) = HP11_613_496501723(3) = HP97_130517_917327(2) = HP53_1832651281459(1) = 3_3_3_11_139_653_3863_5107  (0.014 seconds)
HP9(2) = HP3_3(1) = 3_11  (0.000 seconds)
HP10(4) = HP2_5(3) = HP5_5(2) = HP5_11(1) = 7_73  (0.001 seconds)
HP11 = 11  (0.000 seconds)
HP12(1) = 2_2_3  (0.000 seconds)
HP13 = 13  (0.000 seconds)
HP14(5) = HP2_7(4) = HP3_3_3(3) = HP3_3_37(2) = HP47_71(1) = 13_367  (0.001 seconds)
HP15(4) = HP3_5(3) = HP5_7(2) = HP3_19(1) = 11_29  (0.001 seconds)
HP16(4) = HP2_2_2_2(3) = HP2_11_101(2) = HP3_11_6397(1) = 3_163_6373  (0.001 seconds)
HP17 = 17  (0.000 seconds)
HP18(1) = 2_3_3  (0.000 seconds)
HP19 = 19  (0.000 seconds)
HP20(15) = HP2_2_5(14) = HP3_3_5_5(13) = HP5_11_61(12) = HP11_4651(11) = HP3_3_12739(10) = HP17_194867(9) = HP19_41_22073(8) = HP709_273797(7) = HP3_97_137_17791(6) = HP11_3610337981(5) = HP7_3391_4786213(4) = HP3_3_3_3_7_23_31_1815403(3) = HP13_17_23_655857429041(2) = HP7_7_2688237874641409(1) = 3_31_8308475676071413  (0.020 seconds)
HP65(19) = HP5_13(18) = HP3_3_3_19(17) = HP11_13_233(16) = HP11_101203(15) = HP3_3_23_53629(14) = HP3_3_1523_24247(13) = HP3_3_3_7_47_3732109(12) = HP11_18013_16843763(11) = HP151_740406071813(10) = HP3_13_13_54833_5458223(9) = HP3_3_97_179_373_7523_71411(8) = HP1571_1601_1350675311441(7) = HP3_3_13_33391_143947_279384649(6) = HP11_23_204069263_6417517893491(5) = HP7_11_1756639_83039633268945697(4) = HP29_29_5165653_13503983_12122544283(3) = HP228345060379_1282934064985326977(2) = HP3_3_3_2979253_3030445387_9367290955541(1) = 1381_3211183211_75157763339900357651  (6.686 seconds)
Total elapsed time: 6.737 seconds

HP49:
HP49(n - 0) = 7_7  (0.000 seconds)
HP77(n - 1) = 7_11  (0.000 seconds)
HP711(n - 2) = 3_3_79  (0.000 seconds)
HP3379(n - 3) = 31_109  (0.000 seconds)
HP31109(n - 4) = 13_2393  (0.000 seconds)
HP132393(n - 5) = 3_44131  (0.000 seconds)
HP344131(n - 6) = 17_31_653  (0.000 seconds)
HP1731653(n - 7) = 7_11_43_523  (0.000 seconds)
HP71143523(n - 8) = 11_11_577_1019  (0.000 seconds)
HP11115771019(n - 9) = 311_35742029  (0.000 seconds)
HP31135742029(n - 10) = 7_17_261644891  (0.000 seconds)
HP717261644891(n - 11) = 11_19_3431873899  (0.002 seconds)
HP11193431873899(n - 12) = 11_613_4799_345907  (0.001 seconds)
HP116134799345907(n - 13) = 3_204751_189066719  (0.001 seconds)
HP3204751189066719(n - 14) = 3_1068250396355573  (0.003 seconds)
HP31068250396355573(n - 15) = 621611_49980213343  (0.005 seconds)
HP62161149980213343(n - 16) = 3_3_6906794442245927  (0.006 seconds)
HP336906794442245927(n - 17) = 73_4615161567701999  (0.009 seconds)
HP734615161567701999(n - 18) = 3_13_18836286194043641  (0.009 seconds)
HP31318836286194043641(n - 19) = 3_3_3_43_14369_161461_11627309  (0.004 seconds)
HP333431436916146111627309(n - 20) = 3_32057_1618455677_2142207827  (0.153 seconds)
HP33205716184556772142207827(n - 21) = 3_1367_2221_5573_475297_1376323127  (0.006 seconds)
HP31367222155734752971376323127(n - 22) = 7_3391_51263_25777821480557336017  (0.003 seconds)
HP733915126325777821480557336017(n - 23) = 47_67_347_431_120361987_12947236602187  (0.043 seconds)
HP476734743112036198712947236602187(n - 24) = 3_7_7_17_12809_57470909_57713323_4490256751  (0.124 seconds)
HP377171280957470909577133234490256751(n - 25) = 3096049809383_121823389214993262890297  (27.913 seconds)
HP3096049809383121823389214993262890297(n - 26) = 7_379_62363251_18712936424989555929478399  (0.132 seconds)
HP73796236325118712936424989555929478399(n - 27) = 13_1181_145261411_33089538087518197265265053  (0.034 seconds)
HP13118114526141133089538087518197265265053(n - 28) = 3_19_521_441731977174163487542111577539726749  (0.002 seconds)
HP319521441731977174163487542111577539726749(n - 29) = 59_5415617656474189392601483764603009147911  (0.002 seconds)
HP595415617656474189392601483764603009147911(n - 30) = 13_8423_1466957_3706744784027901056001426046777  (0.015 seconds)```

## Wren

Library: Wren-math
Library: Wren-big
Library: Wren-sort

This uses a combination of the Pollard Rho algorithm and wheel based factorization to try and factorize the large numbers involved here in a reasonable time.

Reaches HP20 in about 0.52 seconds but HP65 took just under 40 minutes!

`import "/math" for Intimport "/big" for BigIntimport "/sort" for Sort // simple wheel based prime factors routine for BigIntvar primeFactorsWheel = Fn.new { |n|    var inc = [4, 2, 4, 2, 4, 6, 2, 6]    var factors = []    while (n%2 == 0) {        factors.add(BigInt.two)        n = n / 2    }    while (n%3 == 0) {        factors.add(BigInt.three)        n = n / 3    }    while (n%5 == 0) {        factors.add(BigInt.five)        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} var pollardRho = Fn.new { |n|    var g = Fn.new { |x, y| (x*x + BigInt.one) % n }    var x = BigInt.two    var y = BigInt.two    var z = BigInt.one    var d = BigInt.one    var count = 0    while (true) {        x = g.call(x, n)        y = g.call(g.call(y, n), n)        d = (x - y).abs % n        z = z * d        count = count + 1        if (count == 100) {            d = BigInt.gcd(z, n)            if (d != BigInt.one) break            z = BigInt.one            count = 0        }    }    if (d == n) return BigInt.zero    return d} var primeFactors = Fn.new { |n|    var factors = []    while (n > 1) {        if (n > BigInt.maxSmall/100) {            var d = pollardRho.call(n)            if (d != 0) {                factors.addAll(primeFactorsWheel.call(d))                n = n / d                if (n.isProbablePrime(2)) {                    factors.add(n)                    break                }            } else {                factors.addAll(primeFactorsWheel.call(n))                break            }        } else {            factors.addAll(primeFactorsWheel.call(n))            break        }    }    Sort.insertion(factors)    return factors} var list = (2..20).toListlist.add(65)for (i in list) {    if (Int.isPrime(i)) {        System.print("HP%(i) = %(i)")        continue    }    var n = 1    var j = BigInt.new(i)    var h = [j]    while (true) {        var k = primeFactors.call(j).reduce("") { |acc, f| acc + f.toString }        j = BigInt.new(k)        h.add(j)        if (j.isProbablePrime(2)) {            for (l in n...0) System.write("HP%(h[n-l])(%(l)) = ")            System.print(h[n])            break        } else {            n = n + 1        }    }}`
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
```