Achilles numbers

This page uses content from Wikipedia. The original article was at Achilles number. 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)

An Achilles number is a number that is powerful but imperfect. Named after Achilles, a hero of the Trojan war, who was also powerful but imperfect.

A positive integer n is a powerful number if, for every prime factor p of n, p² is also a divisor.

In other words, every prime factor appears at least squared in the factorization.

All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as m^k, where m and k are positive integers greater than 1.

A strong Achilles number is an Achilles number whose Euler totient (𝜑) is also an Achilles number.

E.G.

108 is a powerful number. Its prime factorization is 2² × 3³, and thus its prime factors are 2 and 3. Both 2² = 4 and 3² = 9 are divisors of 108. However, 108 cannot be represented as m^k, where m and k are positive integers greater than 1, so 108 is an Achilles number.

360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by 5² = 25.

Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 2² = 4 and 7² = 49 are divisors of it. Nonetheless, it is a perfect power; its square root is an even integer, so it is not an Achilles number.

500 = 2² × 5³ is a strong Achilles number as its Euler totient, 𝜑(500), is 200 = 2³ × 5² which is also an Achilles number.

Task

Find and show the first 50 Achilles numbers.
Find and show at least the first 20 strong Achilles numbers.
For at least 2 through 5, show the count of Achilles numbers with that many digits.

See also

Wikipedia: Achilles number OEIS:A052486 - Achilles numbers - powerful but imperfect numbers OEIS:A194085 - Strong Achilles numbers: Achilles numbers m such that phi(m) is also an Achilles number Related task: Powerful numbers Related task: Totient function

J

Implementation:

<lang J>achilles=: (*/ .>&1 * 1 = +./)@(1{__&q:)"0 strong=: achilles@(5&p:)</lang>

Task examples:

<lang J> 5 10$(#~ achilles) 1+i.10000 NB. first 50 achilles numbers

 72  108  200  288  392  432  500  648  675  800
864  968  972 1125 1152 1323 1352 1372 1568 1800

1944 2000 2312 2592 2700 2888 3087 3200 3267 3456 3528 3872 3888 4000 4232 4500 4563 4608 5000 5292 5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

  20{.(#~ strong * achilles) 1+i.100000 NB. first twenty strong achilles numbers

500 864 1944 2000 2592 3456 5000 10125 10368 12348 12500 16875 19652 19773 30375 31104 32000 33275 37044 40500

  +/achilles (+i.)/1 9*10^<:2  NB. count of two digit achilles numbers

1

  +/achilles (+i.)/1 9*10^<:3

12

  +/achilles (+i.)/1 9*10^<:4

47

  +/achilles (+i.)/1 9*10^<:5

192

  +/achilles (+i.)/1 9*10^<:6

664</lang>

Explanation of the code:

(1{__&q:) is a function which returns the non-zero powers of the prime factors of a positive integer. (__&q: returns both the primes and their factors, but here we do not care about the primes themselves.)

+./ returns the greatest common divisor of a list, and 1=+./ is true if that gcd is 1 (0 if it's false).

*/ .>&1 is true if all the values in a list are greater than 1 (0 if not).

"0 maps a function onto the individual (rank 0) items of a list or array (we use this to avoid complexities: for example if we padded our lists of prime factor powers with zeros, we could still find the gcd, but our test that the powers are greater than 1 would fail).

5&p: is euler's totient function.

(#~ predicate) list selects the elements of list where predicate is true.

Julia

<lang>using Primes

isAchilles(n) = (p = [x[2] for x in factor(n).pe]; all(>(1), p) && gcd(p) == 1)

isstrongAchilles(n) = isAchilles(n) && isAchilles(totient(n))

function teststrongachilles(nachilles = 50, nstrongachilles = 100)

   # task 1
   println("First $nachilles Achilles numbers:")
   n, found = 0, 0
   while found < nachilles
       if isAchilles(n)
           found += 1
           print(rpad(n, 5), found % 10 == 0 ? "\n" : "")
       end
       n += 1
   end
   # task 2
   println("\nFirst $nstrongachilles strong Achilles numbers:")
   n, found = 0, 0
   while found < nstrongachilles
       if isstrongAchilles(n)
           found += 1
           print(rpad(n, 7), found % 10 == 0 ? "\n" : "")
       end
       n += 1
   end
   # task 3
   println("\nCount of Achilles numbers for various intervals:")
   intervals = [10:99, 100:999, 1000:9999, 10000:99999, 100000:999999]
   for interval in intervals
       println(lpad(interval, 15), " ", count(isAchilles, interval))
   end

end

teststrongachilles()

</lang>

Output:

First 50 Achilles numbers:
72   108  200  288  392  432  500  648  675  800
864  968  972  1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 100 strong Achilles numbers:
500    864    1944   2000   2592   3456   5000   10125  10368  12348
12500  16875  19652  19773  30375  31104  32000  33275  37044  40500  
49392  50000  52488  55296  61731  64827  67500  69984  78608  80000  
81000  83349  84375  93312  108000 111132 124416 128000 135000 148176 
151875 158184 162000 165888 172872 177957 197568 200000 202612 209952 
219488 221184 237276 243000 246924 253125 266200 270000 273375 296352 
320000 324000 333396 364500 397953 405000 432000 444528 453789 455877 
493848 497664 500000 518616 533871 540000 555579 583443 605052 607500 
629856 632736 648000 663552 665500 666792 675000 691488 740772 750141 
790272 800000 810448 820125 831875 877952 949104 972000 987696 1000188

Count of Achilles numbers for various intervals:
          10:99 1
        100:999 12
      1000:9999 47
    10000:99999 192
  100000:999999 664

Phix

Library: Phix/online

You can run this online here.

with javascript_semantics 
requires("1.0.2") -- [prime_factors(duplicates:=2) enhancement]
function achilles(integer n, maxprime=-1)
    sequence f = vslice(prime_factors(n,2,maxprime),2)
    return min(f)>=2 and gcd(f)=1
end function

function totient(integer n)
    integer tot = n, i = 2
    while i*i<=n do
        if rmdr(n,i)=0 then
            while true do
                n /= i
                if rmdr(n,i)!=0 then exit end if
            end while
            tot -= tot/i
        end if
        i += iff(i=2?1:2)
    end while
    if n>1 then
        tot -= tot/n
    end if
    return tot
end function

function strong_achilles(integer n, maxprime=-1)
    return achilles(n,maxprime) and achilles(totient(n),maxprime)
end function

atom t0 = time()
integer c = iff(platform()=JS?7:8),
        pn = 0
-- powerful numbers must be multiples of 2 or more squared primes,
-- achilles numbers must be a multiple of at least 1 cubed prime
sequence p2 = repeat(0,power(10,c))
while true do
    pn += 1
    atom pn2 = power(get_prime(pn),2)
    if pn2>length(p2) then exit end if
    for i=pn2 to length(p2) by pn2 do
        p2[i] += 2
    end for
    atom pn3 = power(get_prime(pn),3)
    if pn3<=length(p2) then -- (avoids a for loop limit error...)
        for i=pn3 to length(p2) by pn3 do
            p2[i] = or_bits(p2[i],1)
        end for
    end if
end while
sequence a = {}, sa = {}, counts = repeat(0,c)
bool bPrintDigits = false
for p=1 to c do
    integer p10 = power(10,p-1),
            p99 = power(10,p)-1,
            maxprime = get_maxprime(p99)
    for n=p10 to p99 do
        integer p2n = p2[n]
        if p2n>4 and odd(p2n) and achilles(n,maxprime) then
            if length(a)<50 then
                a = append(a,sprintf("%4d",n))
                if length(a)=50 then
                    printf(1,"First 50 Achilles numbers:\n%s\n",{join_by(a,1,10," ")})
                end if
            end if
            if length(sa)<30 
            and strong_achilles(n,maxprime) then
                sa = append(sa,sprintf("%5d",n))
                if length(sa)=30 then
                    printf(1,"First 30 strong Achilles numbers:\n%s\n",{join_by(sa,1,10," ")})
                    bPrintDigits = true
                end if
            end if
            counts[p] += 1
        end if
    end for
    if bPrintDigits then
        for i=1 to p do
            if counts[i]!=0 then
                printf(1,"Achilles numbers with %d digits:%d\n",{i,counts[i]})
                counts[i] = 0
            end if
        end for
    end if
end for
?elapsed(time()-t0)

Output:

First 50 Achilles numbers:
  72  108  200  288  392  432  500  648  675  800
 864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
  500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Achilles numbers with 2 digits:1
Achilles numbers with 3 digits:12
Achilles numbers with 4 digits:47
Achilles numbers with 5 digits:192
Achilles numbers with 6 digits:664
Achilles numbers with 7 digits:2242
Achilles numbers with 8 digits:7395
"21.2s"

Unfortunately that p2 sieve is a bit of a memory hog and it won't go to 9 digits (didn't bother testing but pretty sure JavaScript would choke on 8).

Raku

Timing is going to be system / OS dependent. <lang perl6>use Prime::Factor; use Math::Root;

sub is-square-free (Int \n) {

   constant @p = ^100 .map: { next unless .is-prime; .² };
   for @p -> \p { return False if n %% p }
   True

}

sub powerful (\n, \k = 2) {

   my @p;
   p(1, 2*k - 1);
   sub p (\m, \r) {
       @p.push(m) and return if r < k;
       for 1 .. (n / m).&root(r) -> \v {
           if r > k {
               next unless is-square-free(v);
               next unless m gcd v == 1;
           }
           p(m * v ** r, r - 1)
       }
   }
   @p

}

my $achilles = powerful(10**9).hyper(:500batch).grep( {

   my $f = .&prime-factors.Bag;
   (+$f.keys > 1) && (1 == [gcd] $f.values) && (.sqrt.Int² !== $_)

} ).classify: { .chars }

my \𝜑 = 0, |(1..*).hyper.map: -> \t { t × [×] t.&prime-factors.squish.map: { 1 - 1/$_ } }

my %as = Set.new: flat $achilles.values».list;

my $strong = lazy (flat $achilles.sort».value».list».sort).grep: { ?%as{𝜑[$_]} };

put "First 50 Achilles numbers:"; put (flat $achilles.sort».value».list».sort)[^50].batch(10)».fmt("%4d").join("\n");

put "\nFirst 30 strong Achilles numbers:"; put $strong[^30].batch(10)».fmt("%5d").join("\n");

put "\nNumber of Achilles numbers with:"; say "$_ digits: " ~ +$achilles{$_} // 0 for 2..9;

printf "\n%.1f total elapsed seconds\n", now - INIT now;</lang>

Output:

First 50 Achilles numbers:
  72  108  200  288  392  432  500  648  675  800
 864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
  500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395
9 digits: 24008

6.1 total elapsed seconds

Could go further but slows to a crawl and starts chewing up memory in short order.

10 digits: 77330
11 digits: 247449
12 digits: 788855

410.4 total elapsed seconds

Wren

Library: Wren-math

Library: Wren-seq

Library: Wren-fmt

Version 1 (Brute force)

This finds the number of 6 digit Achilles numbers in 2.5 seconds, 7 digits in 51 seconds but 8 digits needs a whopping 21 minutes! <lang ecmascript>import "./math" for Int import "./seq" for Lst import "./fmt" for Fmt

var maxDigits = 8 var limit = 10.pow(maxDigits) var c = Int.primeSieve(limit-1, false)

var totient = Fn.new { |n|

   var tot = n
   var i = 2
   while (i*i <= n) {
       if (n%i == 0) {
           while(n%i == 0) n = (n/i).floor
           tot = tot - (tot/i).floor
       }
       if (i == 2) i = 1
       i = i + 2
   }
   if (n > 1) tot = tot - (tot/n).floor
   return tot

}

var isPerfectPower = Fn.new { |n|

   if (n == 1) return true
   var x = 2
   while (x * x <= n) {
       var y = 2
       var p = x.pow(y)
       while (p > 0 && p <= n) {
           if (p == n) return true
           y = y + 1
           p = x.pow(y)
       }
       x = x + 1
   }
   return false

}

var isPowerful = Fn.new { |n|

   while (n % 2 == 0) {
       var p = 0
       while (n % 2 == 0) {
           n = (n/2).floor
           p = p + 1
       }
       if (p == 1) return false
   }
   var f = 3
   while (f * f <= n) {
       var p = 0
       while (n % f == 0) {
           n = (n/f).floor
           p = p + 1
       }
       if (p == 1) return false
       f = f + 2
   }
   return n == 1

}

var isAchilles = Fn.new { |n| c[n] && isPowerful.call(n) && !isPerfectPower.call(n) }

var isStrongAchilles = Fn.new { |n|

   if (!isAchilles.call(n)) return false
   var tot = totient.call(n)
   return isAchilles.call(tot)

}

System.print("First 50 Achilles numbers:") var achilles = [] var count = 0 var n = 2 while (count < 50) {

   if (isAchilles.call(n)) {
       achilles.add(n)
       count = count + 1
   }
   n = n + 1

} for (chunk in Lst.chunks(achilles, 10)) Fmt.print("$4d", chunk)

System.print("\nFirst 30 strong Achilles numbers:") var strongAchilles = [] count = 0 n = achilles[0] while (count < 30) {

   if (isStrongAchilles.call(n)) {
       strongAchilles.add(n)
       count = count + 1
   }
   n = n + 1

} for (chunk in Lst.chunks(strongAchilles, 10)) Fmt.print("$5d", chunk)

System.print("\nNumber of Achilles numbers with:") var pow = 10 for (i in 2..maxDigits) {

   var count = 0
   for (j in pow..pow*10-1) {
       if (isAchilles.call(j)) count = count + 1
   }
   System.print("%(i) digits: %(count)")
   pow = pow * 10

}</lang>

Output:

First 50 Achilles numbers:
  72  108  200  288  392  432  500  648  675  800
 864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
  500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
2 digits: 1
3 digits: 12
4 digits: 47
5 digits: 192
6 digits: 664
7 digits: 2242
8 digits: 7395

Version 2 (Much faster)

Library: Wren-set

Here we make use of the fact that powerful numbers are always of the form a²b³, where a and b > 0, to generate such numbers up to a given limit. We also use an improved method for checking whether a number is a perfect power.

Ridiculously fast compared to the previous method: 8 digits now reached in 0.57 seconds, 9 digits in 4.2 seconds and 10 digits in 34 seconds. However, 11 digits takes 300 seconds and I gave up after that. <lang ecmascript>import "./set" for Set, Bag import "./math" for Int import "./seq" for Lst import "./fmt" for Fmt

var totient = Fn.new { |n|

   var tot = n
   var i = 2
   while (i*i <= n) {
       if (n%i == 0) {
           while(n%i == 0) n = (n/i).floor
           tot = tot - (tot/i).floor
       }
       if (i == 2) i = 1
       i = i + 2
   }
   if (n > 1) tot = tot - (tot/n).floor
   return tot

}

var isPerfectPower = Fn.new { |n|

   if (n == 1) return true
   var pf = Int.primeFactors(n)
   var bag = Bag.new(pf)
   var es = bag.map { |me| me.value }
   if (es.count == 1) return true
   return Int.gcd(es.toList) != 1

}

var getAchilles = Fn.new { |minExp, maxExp|

   var lower = 10.pow(minExp)
   var upper = 10.pow(maxExp)
   var achilles = Set.new() // avoids duplicates
   var count = 0
   for (b in 1..upper.cbrt.floor) {
       var b3 = b * b * b
       for (a in 1..upper.sqrt.floor) {
           var p = b3 * a * a
           if (p >= upper) break
           if (p >= lower) {
               if (!isPerfectPower.call(p)) achilles.add(p)
           }
       }
   }
   return achilles

}

var achillesSet = getAchilles.call(1, 5) // enough for first 2 parts var achilles = achillesSet.toList achilles.sort()

System.print("First 50 Achilles numbers:") for (chunk in Lst.chunks(achilles[0..49], 10)) Fmt.print("$4d", chunk)

System.print("\nFirst 30 strong Achilles numbers:") var strongAchilles = [] var count = 0 var n = 0 while (count < 30) {

   var tot = totient.call(achilles[n])
   if (achillesSet.contains(tot)) {
       strongAchilles.add(achilles[n])
       count = count + 1
   }
   n = n + 1

} for (chunk in Lst.chunks(strongAchilles, 10)) Fmt.print("$5d", chunk)

var maxDigits = 11 System.print("\nNumber of Achilles numbers with:") for (d in 2..maxDigits) {

   var ac = getAchilles.call(d-1, d).count
   Fmt.print("$2d digits: $d", d, ac)

}</lang>

Output:

First 50 Achilles numbers:
  72  108  200  288  392  432  500  648  675  800
 864  968  972 1125 1152 1323 1352 1372 1568 1800
1944 2000 2312 2592 2700 2888 3087 3200 3267 3456
3528 3872 3888 4000 4232 4500 4563 4608 5000 5292
5324 5400 5408 5488 6075 6125 6272 6728 6912 7200

First 30 strong Achilles numbers:
  500   864  1944  2000  2592  3456  5000 10125 10368 12348
12500 16875 19652 19773 30375 31104 32000 33275 37044 40500
49392 50000 52488 55296 61731 64827 67500 69984 78608 80000

Number of Achilles numbers with:
 2 digits: 1
 3 digits: 12
 4 digits: 47
 5 digits: 192
 6 digits: 664
 7 digits: 2242
 8 digits: 7395
 9 digits: 24008
10 digits: 77330
11 digits: 247449