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A082612 Numbers n such that ((n-1)^2+1)/2 and n^2+1 and ((n+1)^2+1)/2 are prime if n is even or (n-1)^2+1 and (n^2+1)/2 and (n+1)^2+1 are prime if n is odd.

%I A082612 #9 Mar 19 2018 19:04:42
%S A082612 3,4,5,10,15,25,170,205,570,715,780,950,1095,1315,1420,1615,2055,2380,
%T A082612 2405,2730,2925,3755,3850,4120,4300,4615,4795,5015,5055,5475,5850,
%U A082612 6360,6460,6785,6800,6970,7100,7240,7855,8115,8175,8720,9425,9475,9630,10150
%N A082612 Numbers n such that ((n-1)^2+1)/2 and n^2+1 and ((n+1)^2+1)/2 are prime if n is even or (n-1)^2+1 and (n^2+1)/2 and (n+1)^2+1 are prime if n is odd.
%C A082612 I believe this is an infinite sequence, though a proof seems to be still far off. 155th term is 62910. There are probably infinitely many consecutive n^2+1 or (n^2+1)/2 primes. That is, n^2+1 and (n+2)^2+1 or (n^2+1)/2 and ((n+2)^2+1)/2 are both prime infinitely often.
%H A082612 Harvey P. Dale, <a href="/A082612/b082612.txt">Table of n, a(n) for n = 1..1000</a>
%e A082612 a(4)=10 (9^2+1)/2=41 and 10^2+1=101 and (11^2+1)/2=61 are prime.
%t A082612 neoQ[n_]:=If[EvenQ[n],AllTrue[{((n-1)^2+1)/2,n^2+1,((n+1)^2+1)/2}, PrimeQ], AllTrue[{(n-1)^2+1, (n^2+1)/2,(n+1)^2+1},PrimeQ]]; Select[Range[ 6400], neoQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Mar 19 2018 *)
%K A082612 nonn
%O A082612 1,1
%A A082612 _Robin Garcia_, Sep 23 2004
%E A082612 More terms from _Harvey P. Dale_, Mar 19 2018