A370387 a(n) is the least prime p such that p + 6*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

2, 19, 5, 67, 7, 281, 1051, 6791, 11, 115599457, 365705201, 79352440891, 286351937491, 5810592517241, 17, 1942721697854617

Offset: 1

Author

J.W.L. (Jan) Eerland, Mar 12 2024

Keywords

• nonn
• more

Comments

a(10), ..., a(14) > 10^7, a(15) = 17, a(16), ..., a(20) > 10^7.

a(29) = 31. - Chai Wah Wu, Apr 10 2024

Crossrefs

Cf. A164926, A371024.

Programs

Maple

f:= proc(p) local k;
  for k from 1 while isprime(p+k*(k+1)*6) do od:
  k
end proc:
A:= Vector(12): count:= 0:
for i from 1 while count < 12 do
  v:= f(ithprime(i));
  if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi
od:
convert(A,list);

Mathematica

Table[p=1;m=6;Monitor[Parallelize[While[True,If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1),{k,0,n-1}]],False]==False,PrimeQ[p+m*n*(n+1)]==False],Break[]];p++];p],p],{n,1,10}]

PARI

isok(p, n) = for (k=0, n-1, if (! isprime(p + 6*k*(k+1)), return(0))); return (!isprime(p + 6*n*(n+1)));
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p;

Perl

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 6*$n*($n+1)) } sieve_prime_cluster($lo, $hi, map { 6*$*($+1) } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " } # Daniel Suteu, Dec 30 2024

Extensions

a(10)-a(11) from Chai Wah Wu, Apr 10 2024

a(12) from Chai Wah Wu, Apr 11 2024

a(13)-a(14) from David A. Corneth, Apr 11 2024

a(15) from J.W.L. (Jan) Eerland, Mar 12 2024

a(16) from Daniel Suteu, Dec 30 2024