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A359396 a(n) is the least k such that k^j+2 is prime for j = 1 to n but not n+1.

%I A359396 #15 Jan 11 2023 08:48:08
%S A359396 5,9,105,3,909,4995825,28212939
%N A359396 a(n) is the least k such that k^j+2 is prime for j = 1 to n but not n+1.
%C A359396 All terms are odd, and all except a(1) = 5 are divisible by 3.
%C A359396 The generalized Bunyakovsky conjecture implies that a(n) exists for all n.
%C A359396 a(8) > 10^10.
%C A359396 a(8) > 10^11. - _Lucas A. Brown_, Jan 11 2023
%e A359396 a(4) = 3 because 3^1 + 2 = 5, 3^2 + 2 = 11, and 3^3 + 2 = 29 and 3^4 + 2 = 83 are prime but 3^5 + 2 = 245 is not.
%p A359396 f:= proc(n) local j;
%p A359396  for j from 1 do
%p A359396      if not isprime(n^j+2) then return j-1 fi
%p A359396  od
%p A359396 end proc:
%p A359396 V:= Vector(7): V[1]:= 5: count:= 1:
%p A359396 for k from 3 by 6 while count < 7 do
%p A359396  v:= f(k);
%p A359396  if v > 0 and V[v] = 0 then V[v]:= k; count:= count+1 fi
%p A359396 od:
%p A359396 convert(V,list);
%o A359396 (Python)
%o A359396 from sympy import isprime
%o A359396 from itertools import count, islice
%o A359396 def f(k):
%o A359396     j = 1
%o A359396     while isprime(k**j + 2): j += 1
%o A359396     return j-1
%o A359396 def agen():
%o A359396     adict, n = dict(), 1
%o A359396     for k in count(2):
%o A359396         v = f(k)
%o A359396         if v not in adict: adict[v] = k
%o A359396         while n in adict: yield adict[n]; n += 1
%o A359396 print(list(islice(agen(), 5))) # _Michael S. Branicky_, Jan 09 2023
%Y A359396 Cf. A087576.
%K A359396 nonn,more
%O A359396 1,1
%A A359396 _Robert Israel_, Dec 29 2022