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A378839 a(n) is the least prime p such that p + 8*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

%I A378839 #13 Dec 18 2024 16:41:36
%S A378839 2,3,151,181,13,811,23671,92221,45417481,5078503,4861,20379346831,
%T A378839 12180447943,31,10347699089473
%N A378839 a(n) is the least prime p such that p + 8*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.
%C A378839 No further terms < 2.5*10^11. - _Michael S. Branicky_, Dec 16 2024
%p A378839 f:= proc(p) local k;
%p A378839   for k from 1 while isprime(p+k*(k+1)*8) do od:
%p A378839   k
%p A378839 end proc:
%p A378839 A:= Vector(12): count:= 0:
%p A378839 for i from 1 while count < 12 do
%p A378839   v:= f(ithprime(i));
%p A378839   if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi
%p A378839 od:
%p A378839 convert(A,list);
%t A378839 Table[p=1;m=8;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 A378839 (PARI) isok(p, n) = for (k=0, n-1, if (! isprime(p + 8*k*(k+1)), return(0))); return (!isprime(p + 8*n*(n+1)));
%o A378839 a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p;
%o A378839 (Perl) use ntheory qw(:all); sub a { my $n = $_[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($_ + 8*$n*($n+1)) } sieve_prime_cluster($lo, $hi, map { 8*$_*($_+1) } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " }; #
%Y A378839 Cf. A164926, A370387, A371024, A376675.
%K A378839 nonn,more
%O A378839 1,1
%A A378839 _J.W.L. (Jan) Eerland_, Dec 09 2024
%E A378839 a(12)-a(14) from _Michael S. Branicky_, Dec 15 2024
%E A378839 a(15) from _Daniel Suteu_, Dec 17 2024