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A375912 Primes p such that p*nextprime(p)+1 and p + nextprime(p)+1 are both perfect squares where nextprime(p) is the smallest prime that is larger than p.

%I A375912 #11 Sep 13 2024 07:44:38
%S A375912 3,11,59,179,311,419,2111,3119,5099,21011,21839,24419,30011,37811,
%T A375912 41759,44699,60899,68819,83639,86111,100799,135719,143111,161879,
%U A375912 163019,165311,177011,210599,218459,241511,273059,304979,312839,437111,450299,491039,584279,595139,603899,637319
%N A375912 Primes p such that p*nextprime(p)+1 and p + nextprime(p)+1 are both perfect squares where nextprime(p) is the smallest prime that is larger than p.
%e A375912 11 is a term because 11*nextprime(11)+1 = 12^2 and 11 + nextprime(11)+1 = 5^2.
%p A375912 nn:=10^5:
%p A375912 for n from 1 to nn do:
%p A375912 p:=ithprime(n):q:=nextprime(p):p1:=sqrt(p*q+1):p2:=sqrt(q+p+1):
%p A375912  if floor(p1) = p1 and floor(p2)=p2
%p A375912   then
%p A375912   printf(`%d, `,p):
%p A375912   else
%p A375912  fi:
%p A375912 od:
%t A375912 Select[Partition[Prime[Range[100000]], 2, 1], IntegerQ[Sqrt[#[[1]] + #[[2]] + 1]] && IntegerQ[Sqrt[#[[1]]*#[[2]] + 1]] &][[;; , 1]] (* _Amiram Eldar_, Sep 02 2024 *)
%Y A375912 Intersection of A001359 and A242384.
%K A375912 nonn
%O A375912 1,1
%A A375912 _Michel Lagneau_, Sep 02 2024