A370387 - OEIS

%I #60 Dec 30 2024 13:17:11

%S 2,19,5,67,7,281,1051,6791,11,115599457,365705201,79352440891,

%T 286351937491,5810592517241,17,1942721697854617

%N 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.

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

%C a(29) = 31. - _Chai Wah Wu_, Apr 10 2024

%p f:= proc(p) local k;

%p   for k from 1 while isprime(p+k*(k+1)*6) do od:

%p   k

%p end proc:

%p A:= Vector(12): count:= 0:

%p for i from 1 while count < 12 do

%p   v:= f(ithprime(i));

%p   if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi

%p od:

%p convert(A,list);

%t 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}]

%o (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)));

%o a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p;

%o (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

%Y Cf. A164926, A371024.

%K nonn,more

%O 1,1

%A _J.W.L. (Jan) Eerland_, Mar 12 2024

%E a(10)-a(11) from _Chai Wah Wu_, Apr 10 2024

%E a(12) from _Chai Wah Wu_, Apr 11 2024

%E a(13)-a(14) from _David A. Corneth_, Apr 11 2024

%E a(15) from _J.W.L. (Jan) Eerland_, Mar 12 2024

%E a(16) from _Daniel Suteu_, Dec 30 2024