Penta-power prime seeds

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

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.

   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 quad-power plus 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 @qppps = 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 quad-power plus prime seeds:\n" ~ @qppps[^30].batch(10)».&comma».fmt("%9s").join: "\n";

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

for 1..10 {

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

}</lang>

Output:

First thirty quad-power plus 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 quad-power plus 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