Super-Poulet numbers

A super-Poulet number is a Poulet number (or Fermat pseudoprime to base 2) whose every divisor d evenly divides 2^d − 2.

Task

Find and display the first 20 super-Poulet numbers.

Stretch

Find and display the index and value of the first super-Poulet number greater than one million.

See also

Task: Fermat pseudoprimes Wikipedia: super-Poulet number OEIS:A050217 - super-Poulet numbers

Factor

Works with: Factor version 0.99 2022-04-03

USING: combinators.short-circuit io kernel lists lists.lazy math
math.functions math.primes math.primes.factors prettyprint
sequences ;

: super-poulet? ( n -- ? )
    {
        [ prime? not ]
        [ [ 1 - 2^ ] keep mod 1 = ]
        [ divisors [ [ 2^ 2 - ] keep divisor? ] all? ]
    } 1&& ;

: super-poulets ( -- list )
    1 lfrom [ super-poulet? ] lfilter ;

20 super-poulets ltake [ pprint bl ] leach nl

Output:

341 1387 2047 2701 3277 4033 4369 4681 5461 7957 8321 10261 13747 14491 15709 18721 19951 23377 31417 31609

J

Implementation (only good for the first 60 super-poulet numbers):

spou1=: {{ 2 = 2x(y&|)@^ y }}

is_super_poulet=: {{
   if. 2~:#q=. q: y do. 0 return. end.
   if. spou1 {. q do.
     if. spou1 {: q do.
       if. spou1 y do. 1 return. end.
     end.
   end.
   0
}}"0

Task example:

   20{. (#~ is_super_poulet) 1+i.1e5
341 1387 2047 2701 3277 4033 4369 4681 5461 7957 8321 10261 13747 14491 15709 18721 19951 23377 31417 31609

Julia

using Primes

divisors(n) = @view sort!(vec(map(prod, Iterators.product((p.^(0:m) for (p, m) in eachfactor(n))...))))[begin:end-1]

issuperPoulet(n) = !isprime(n) && big"2"^(n - 1) % n == 1 && all(d -> (big"2"^d - 2) % d == 0, divisors(n))

spoulets = filter(issuperPoulet, 1:12_000_000)

println("The first 20 super-Poulet numbers are: ", spoulets[1:20])

idx1m = findfirst(>(1_000_000), spoulets)
idx10m = findfirst(>(10_000_000), spoulets)

println("The first super-Poulet number over 1 million is the ", idx1m, "th one, which is ", spoulets[idx1m])
println("The first super-Poulet number over 10 million is the ", idx10m, "th one, which is ", spoulets[idx10m])

Output:

The first 20 super-Poulet numbers are: [341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609]
The first super-Poulet number over 1 million is the 109th one, which is 1016801
The first super-Poulet number over 10 million is the 317th one, which is 10031653

Phix

with javascript_semantics
include mpfr.e 
constant two = mpz_init(2)

function super_poulet(integer x)
    if is_prime(x) or x<=1 then return false end if
    mpz z = mpz_init(x)
    mpz_powm_ui(z, two, x-1, z)
    if mpz_cmp_si(z,1)!=0 then return false end if
    sequence f = factors(x)
    for i=1 to length(f) do
        mpz_set_si(z,f[i])
        mpz_powm(z, two, z, z)
        if mpz_cmp_si(z,2)!=0 then return false end if
    end for
    return true
end function
 
integer count = 0, x = 1
sequence first20 = {}
while count<20 do
    if super_poulet(x) then
        if count<20 then first20 &= x end if
        count += 1
    end if
    x += 2
end while
printf(1,"First 20 super-Poulet numbers:\n%v\n\n",{first20})
for lim in {1e6,1e7} do
    while true do
        x += 2
        if super_poulet(x) then
            count += 1
            if x>lim then exit end if
        end if
    end while
    printf(1,"The %d%s super-Poulet number is %,d and the first greater than %,d\n",
             {count,ord(count),x,lim})
end for

Output:

First 20 super-Poulet numbers:
{341,1387,2047,2701,3277,4033,4369,4681,5461,7957,8321,10261,13747,14491,15709,18721,19951,23377,31417,31609}

The 109th super-Poulet number is 1,016,801 and the first greater than 1,000,000
The 317th super-Poulet number is 10,031,653 and the first greater than 10,000,000

Python

Translation of: Julia

from sympy import isprime, divisors
 
def is_super_Poulet(n):
    return not isprime(n) and 2**(n - 1) % n == 1 and all((2**d - 2) % d == 0 for d in divisors(n))

spoulets = [n for n in range(1, 1_100_000) if is_super_Poulet(n)]

print('The first 20 super-Poulet numbers are:', spoulets[:20])

idx1m, val1m = next((i, v) for i, v in enumerate(spoulets) if v > 1_000_000)
print(f'The first super-Poulet number over 1 million is the {idx1m}th one, which is {val1m}')

Output:

The first 20 super-Poulet numbers are: [341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609]
The first super-Poulet number over 1 million is the 108th one, which is 1016801

Raku

use Prime::Factor;
use Lingua::EN::Numbers;

my @poulet = lazy (2..*).hyper(:2000batch).grep: { !.is-prime && (1 == expmod 2, $_ - 1, $_) }
my @super-poulet = @poulet.grep: { all .&divisors.skip(1).map: { 2 == expmod 2, $_, $_ } }

say "First 20 super-Poulet numbers:\n" ~ @super-poulet[^20].gist;

for 1e6.Int, 1e7.Int -> $threshold {
    say "\nIndex and value of first super-Poulet greater than {$threshold.&cardinal}:";
    my $index = @super-poulet.first: * > $threshold, :k;
    say "{(1+$index).&ordinal-digit} super-Poulet number == " ~ @super-poulet[$index].&comma;
}

Output:

First 20 super-Poulet numbers:
(341 1387 2047 2701 3277 4033 4369 4681 5461 7957 8321 10261 13747 14491 15709 18721 19951 23377 31417 31609)

Index and value of first super-Poulet greater than one million:
109th super-Poulet number == 1,016,801

Index and value of first super-Poulet greater than ten million:
317th super-Poulet number == 10,031,653

Wren

Library: Wren-math

Library: Wren-gmp

Library: Wren-fmt

import "./math" for Int
import "./gmp" for Mpz
import "./fmt" for Fmt

var one = Mpz.one

var isSuperPoulet = Fn.new { |x|
    if (Int.isPrime(x)) return false
    var bx = Mpz.from(x)
    if (Mpz.two.modPow(x-1, bx) != one) return false
    var t = Mpz.new()
    return Int.divisors(x).skip(1).all { |d| t.uiPow(2, d).sub(2).isDivisibleUi(d) }
}

var count = 0
var first20 = List.filled(20, 0)
var x = 3
while (count < 20) {
    if (isSuperPoulet.call(x)) {
        first20[count] = x
        count = count + 1
    }
    x = x + 2  // Poulet numbers are always odd
}
System.print("The first 20 super-Poulet numbers are:")
System.print(first20)
System.print()
var limit = 1e6
while (true) {
   if (isSuperPoulet.call(x)) {
        count = count + 1
        if (x > limit) {
            Fmt.print("The $r super-Poulet number and the first over $,d is $,d.", count, limit, x)
            if (limit == 1e6) limit = 1e7 else return
        }
   }
   x = x + 2
}

Output:

The first 20 super-Poulet numbers are:
[341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 14491, 15709, 18721, 19951, 23377, 31417, 31609]

The 109th super-Poulet number and the first over 1,000,000 is 1,016,801.
The 317th super-Poulet number and the first over 10,000,000 is 10,031,653.