Home primes: Difference between revisions
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Revision as of 15:01, 25 May 2021
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).
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.
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