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.
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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), for instance HP4(2) === HP22(1) === 211 is the same as saying home_prime(4) needs 2 iterations and is the same as home_prime(22) which needs 1 iteration, and (both) equal 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
<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 ;
- 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</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
Julia
<lang 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)
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
</lang>
- 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.
Phix
No stretch goal here this time, just the basic task.
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[$][1] mpz p = mpz_init(lastp) if not mpz_prime(p) or length(rr)=1 then exit end if rr = rr[1..$-1] lastp = "_"&lastp elsif length(rr)=1 and rr[1][2]=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
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 $step = +@steps;
say +@steps > 1
?? (@steps[0..*-2].map( { "HP$_\({--$step})" } ).join: ' = ')
!! ("HP$m"), " = ", @steps[*-1];
}</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
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! <lang ecmascript>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
}
}
}</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