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Home primes

From Rosetta Code
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[edit]

Works with: Factor version 0.99 2021-02-05
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 ;
 
: 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

Go[edit]

Translation of: Wren
Library: Go-rcu
package main
 
import (
"fmt"
"math/big"
"rcu"
"sort"
)
 
var zero = new(big.Int)
var one = big.NewInt(1)
var two = big.NewInt(2)
var three = big.NewInt(3)
var four = big.NewInt(4)
var five = big.NewInt(5)
var six = big.NewInt(6)
 
// simple wheel based prime factors routine for BigInt
func primeFactorsWheel(m *big.Int) []*big.Int {
n := new(big.Int).Set(m)
t := new(big.Int)
inc := []*big.Int{four, two, four, two, four, six, two, six}
var factors []*big.Int
for t.Rem(n, two).Cmp(zero) == 0 {
factors = append(factors, two)
n.Quo(n, two)
}
for t.Rem(n, three).Cmp(zero) == 0 {
factors = append(factors, three)
n.Quo(n, three)
}
for t.Rem(n, five).Cmp(zero) == 0 {
factors = append(factors, five)
n.Quo(n, five)
}
k := big.NewInt(7)
i := 0
for t.Mul(k, k).Cmp(n) <= 0 {
if t.Rem(n, k).Cmp(zero) == 0 {
factors = append(factors, new(big.Int).Set(k))
n.Quo(n, k)
} else {
k.Add(k, inc[i])
i = (i + 1) % 8
}
}
if n.Cmp(one) > 0 {
factors = append(factors, n)
}
return factors
}
 
func pollardRho(n *big.Int) *big.Int {
g := func(x, n *big.Int) *big.Int {
x2 := new(big.Int)
x2.Mul(x, x)
x2.Add(x2, one)
return x2.Mod(x2, n)
}
x, y, d := new(big.Int).Set(two), new(big.Int).Set(two), new(big.Int).Set(one)
t, z := new(big.Int), new(big.Int).Set(one)
count := 0
for {
x = g(x, n)
y = g(g(y, n), n)
t.Sub(x, y)
t.Abs(t)
t.Mod(t, n)
z.Mul(z, t)
count++
if count == 100 {
d.GCD(nil, nil, z, n)
if d.Cmp(one) != 0 {
break
}
z.Set(one)
count = 0
}
}
if d.Cmp(n) == 0 {
return new(big.Int)
}
return d
}
 
func primeFactors(m *big.Int) []*big.Int {
n := new(big.Int).Set(m)
var factors []*big.Int
lim := big.NewInt(1e9)
for n.Cmp(one) > 0 {
if n.Cmp(lim) > 0 {
d := pollardRho(n)
if d.Cmp(zero) != 0 {
factors = append(factors, primeFactorsWheel(d)...)
n.Quo(n, d)
if n.ProbablyPrime(10) {
factors = append(factors, n)
break
}
} else {
factors = append(factors, primeFactorsWheel(n)...)
break
}
} else {
factors = append(factors, primeFactorsWheel(n)...)
break
}
}
sort.Slice(factors, func(i, j int) bool { return factors[i].Cmp(factors[j]) < 0 })
return factors
}
 
func main() {
list := make([]int, 20)
for i := 2; i <= 20; i++ {
list[i-2] = i
}
list[19] = 65
for _, i := range list {
if rcu.IsPrime(i) {
fmt.Printf("HP%d = %d\n", i, i)
continue
}
n := 1
j := big.NewInt(int64(i))
h := []*big.Int{j}
for {
pf := primeFactors(j)
k := ""
for _, f := range pf {
k += fmt.Sprintf("%d", f)
}
j, _ = new(big.Int).SetString(k, 10)
h = append(h, j)
if j.ProbablyPrime(10) {
for l := n; l > 0; l-- {
fmt.Printf("HP%d(%d) = ", h[n-l], l)
}
fmt.Println(h[n])
break
} else {
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

J[edit]

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.21
exit 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[edit]

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)
end
homeprimechain(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")
end
end
 
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[edit]

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, strutils
import 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) & 65
for 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[edit]

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[edit]

Added a new mpz_pollard_rho routine, based on the wp:Pollard's_rho_algorithm C code.

with javascript_semantics
requires("1.0.0")
include mpfr.e

procedure test(integer n)
    string s = sprintf("%d",n), lastp = ""
    sequence res = {s}
    atom t0 = time()
    while true do
        s = substitute(s,"_","")
        sequence rr = mpz_pollard_rho(s,true)
        if length(rr)=1 then exit end if
        s = join(rr,"_")
        res = append(res,s)
    end while
    atom t = time()-t0
    integer niter = length(res)-1
    string iter = iff(niter>1?sprintf("(%d)",niter):""),
           e = iff(t>0.1?" ["&elapsed(t)&"]":"")
    s = sprintf("HP%d%s = ",{n,iter})
    if niter=0 then
        printf(1,"%s%d %s\n",{s,n,e})
    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 %s\n",{s,res[$],e})
    end if
end procedure
papply(tagset(20,2)&65,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
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  [13.9s]

Raku[edit]

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)

REXX[edit]

/*REXX program finds and displays the   home prime   of a range of positive integers.   */
numeric digits 20 /*ensure handling of larger integers. */
parse arg LO HI . /*obtain optional arguments from the CL*/
if LO=='' | LO=="," then LO= 2 /*Not specified? Then use the default.*/
if HI=='' | HI=="," then HI= 20 /* " " " " " " */
@hpc= 'home prime chain for ' /*a literal used in two SAY statements.*/
w= length(HI) /*HI width, used for output alignment. */
do j=max(2, LO) to HI /*find home primes for an integer range*/
pf= factr(j); f= words(pf) /*get prime factors; number of factors.*/
if f==1 then do; say @hpc j": " j ' is prime.'; iterate; end /*J is prime*/
xxx.1= j /*save J in the first array element. */
do n=2 until #==1 /*keep processing until we find a prime*/
xxx.n= space(pf, 0) /*obtain factors of a concatenated p.f.*/
pf= factr(xxx.n); #= words(pf) /*assign factors to PF; # of factors. */
end /*n*/
ee= n /*save EE as the final (last) prime. */
n= n - 1; z= n /*adjust N (for DO loop); assign N to Z*/
$= /*nullify the string of home primes. */
do m=1 for n /*build a list ($) of " " */
$= $ 'HP'xxx.m"("z') ─► ' /*concatenate to string of " " */
z= z - 1 /*decrease the index counter by unity. */
end /*m*/ /* [↑] the index counter is decreasing*/
 
say @hpc right(j, w)":" $ xxx.ee ' is prime.' /*show string of home primes.*/
end /*n*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
factr: procedure; parse arg x 1 d,$ /*set X, D to argument 1; $ to null.*/
if x==1 then return '' /*handle the special case of X = 1. */
do while x//2==0; $= $ 2; x= x% 2; end /*append all the 2 factors of new X.*/
do while x//3==0; $= $ 3; x= x% 3; end /* " " " 3 " " " " */
do while x//5==0; $= $ 5; x= x% 5; end /* " " " 5 " " " " */
do while x//7==0; $= $ 7; x= x% 7; end /* " " " 7 " " " " */
q= 1; r= 0 /*R: will be iSqrt(x). ___*/
do while q<=x; q=q*4; end /*these two lines compute integer √ X */
do while q>1; q=q%4; _= d-r-q; r= r%2; if _>=0 then do; d= _; r= r+q; end; end
 
do k=11 by 6 to r /*insure that J isn't divisible by 3.*/
parse var k '' -1 _ /*obtain the last decimal digit of K. */
if _\==5 then do while x//k==0; $=$ k; x=x%k; end /*maybe reduce by K.*/
if _ ==3 then iterate /*Is next Y is divisible by 5? Skip.*/
y= k+2; do while x//y==0; $=$ y; x=x%y; end /*maybe reduce by Y.*/
end /*k*/
/* [↓] The $ list has a leading blank.*/
if x==1 then return $ /*Is residual=unity? Then don't append.*/
return $ x /*return $ with appended residual. */
output   when using the default input:
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.

Sidef[edit]

for n in (2..20, 65) {
 
var steps = []
var orig = n
 
for (var f = n.factor; true; f = n.factor) {
steps << f
n = Num(f.join)
break if n.is_prime
}
 
say ("HP(#{orig}) = ", steps.map { .join('_') }.join(' -> '))
}
Output:
HP(2) = 2
HP(3) = 3
HP(4) = 2_2 -> 2_11
HP(5) = 5
HP(6) = 2_3
HP(7) = 7
HP(8) = 2_2_2 -> 2_3_37 -> 3_19_41 -> 3_3_3_7_13_13 -> 3_11123771 -> 7_149_317_941 -> 229_31219729 -> 11_2084656339 -> 3_347_911_118189 -> 11_613_496501723 -> 97_130517_917327 -> 53_1832651281459 -> 3_3_3_11_139_653_3863_5107
HP(9) = 3_3 -> 3_11
HP(10) = 2_5 -> 5_5 -> 5_11 -> 7_73
HP(11) = 11
HP(12) = 2_2_3
HP(13) = 13
HP(14) = 2_7 -> 3_3_3 -> 3_3_37 -> 47_71 -> 13_367
HP(15) = 3_5 -> 5_7 -> 3_19 -> 11_29
HP(16) = 2_2_2_2 -> 2_11_101 -> 3_11_6397 -> 3_163_6373
HP(17) = 17
HP(18) = 2_3_3
HP(19) = 19
HP(20) = 2_2_5 -> 3_3_5_5 -> 5_11_61 -> 11_4651 -> 3_3_12739 -> 17_194867 -> 19_41_22073 -> 709_273797 -> 3_97_137_17791 -> 11_3610337981 -> 7_3391_4786213 -> 3_3_3_3_7_23_31_1815403 -> 13_17_23_655857429041 -> 7_7_2688237874641409 -> 3_31_8308475676071413
HP(65) = 5_13 -> 3_3_3_19 -> 11_13_233 -> 11_101203 -> 3_3_23_53629 -> 3_3_1523_24247 -> 3_3_3_7_47_3732109 -> 11_18013_16843763 -> 151_740406071813 -> 3_13_13_54833_5458223 -> 3_3_97_179_373_7523_71411 -> 1571_1601_1350675311441 -> 3_3_13_33391_143947_279384649 -> 11_23_204069263_6417517893491 -> 7_11_1756639_83039633268945697 -> 29_29_5165653_13503983_12122544283 -> 228345060379_1282934064985326977 -> 3_3_3_2979253_3030445387_9367290955541 -> 1381_3211183211_75157763339900357651

Wren[edit]

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 Int
import "/big" for BigInt
import "/sort" for Sort
 
// simple wheel based prime factors routine for BigInt
var 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).toList
list.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