Penta-power prime seeds

Penta-power prime seeds 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.

Generate the sequence of penta-power prime seeds: positive integers n such that:

   n0 + n + 1, n1 + n + 1, n2 + n + 1, n3 + n + 1 and n4 + n + 1 are all prime.

Task

• Find and display the first thirty penta-power prime seeds.

Stretch

• Find and display the position and value of first with a value greater than ten million.

See also

• Task: Quad-power prime seeds

I can find no mention or record of this sequence anywhere. Perhaps I've invented it.

Raku

<lang perl6>use Lingua::EN::Numbers;

my @ppps = lazy (1..*).hyper(:2000batch).grep: -> \n { my \k = n + 1; (1+k).is-prime && (n+k).is-prime && (n²+k).is-prime && (n³+k).is-prime && (n⁴+k).is-prime }

say "First thirty penta-power prime seeds:\n" ~ @ppps[^30].batch(10)».&comma».fmt("%9s").join: "\n";

say "\nFirst penta-power prime seed greater than:";

for 1..10 {

   my $threshold = Int(1e6 * $_);
   my $key = @ppps.first: * > $threshold, :k;
   say "{$threshold.&cardinal.fmt: '%13s'} is the {ordinal-digit $key + 1}: {@ppps[$key].&comma}";

}</lang>

Output:

First thirty penta-power prime seeds:
        1         5        69     1,665     2,129    25,739    29,631    62,321    77,685    80,535
   82,655   126,489   207,285   211,091   234,359   256,719   366,675   407,945   414,099   628,859
  644,399   770,531   781,109   782,781   923,405 1,121,189 1,158,975 1,483,691 1,490,475 1,512,321

First penta-power prime seed greater than:
  one million is the 26th: 1,121,189
  two million is the 39th: 2,066,079
three million is the 47th: 3,127,011
 four million is the 51st: 4,059,525
 five million is the 59th: 5,279,175
  six million is the 63rd: 6,320,601
seven million is the 68th: 7,291,361
eight million is the 69th: 8,334,915
 nine million is the 71st: 9,100,671
  ten million is the 72nd: 10,347,035

Wren

Library: Wren-gmp

Library: Wren-fmt

<lang ecmascript>import "./gmp" for Mpz import "./fmt" for Fmt

var p = Mpz.new() var q = Mpz.one

var isPentaPowerPrimeSeed = Fn.new { |n|

   p.setUi(n)
   var k = n + 1
   return (q + k).probPrime(15) > 0 &&
          (p + k).probPrime(15) > 0 &&
          (p.mul(n) + k).probPrime(15) > 0 &&
          (p.mul(n) + k).probPrime(15) > 0 &&
          (p.mul(n) + k).probPrime(15) > 0

}

var ppps = [] var n = 1 while (ppps.count < 30) {

   if (isPentaPowerPrimeSeed.call(n)) ppps.add(n)
   n = n + 2  // n must be odd

} System.print("First thirty penta-power prime seeds:") Fmt.tprint("$,9d", ppps, 10)

System.print("\nFirst penta-power prime seed greater than:") n = 1 var m = 1 var c = 0 while (true) {

   if (isPentaPowerPrimeSeed.call(n)) {
       c = c + 1
       if (n > m * 1e6) {
           Fmt.print(" $2d million is the $r: $,10d", m, c, n)
           m = m + 1
           if (m == 11) return
       }
   }
   n = n + 2

}</lang>

Output:

First thirty penta-power prime seeds:
        1         5        69     1,665     2,129    25,739    29,631    62,321    77,685    80,535 
   82,655   126,489   207,285   211,091   234,359   256,719   366,675   407,945   414,099   628,859 
  644,399   770,531   781,109   782,781   923,405 1,121,189 1,158,975 1,483,691 1,490,475 1,512,321 

First penta-power prime seed greater than:
  1 million is the 26th:  1,121,189
  2 million is the 39th:  2,066,079
  3 million is the 47th:  3,127,011
  4 million is the 51st:  4,059,525
  5 million is the 59th:  5,279,175
  6 million is the 63rd:  6,320,601
  7 million is the 68th:  7,291,361
  8 million is the 69th:  8,334,915
  9 million is the 71st:  9,100,671
 10 million is the 72nd: 10,347,035