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.

3, 4, 5, 10, 15, 25, 170, 205, 570, 715, 780, 950, 1095, 1315, 1420, 1615, 2055, 2380, 2405, 2730, 2925, 3755, 3850, 4120, 4300, 4615, 4795, 5015, 5055, 5475, 5850, 6360, 6460, 6785, 6800, 6970, 7100, 7240, 7855, 8115, 8175, 8720, 9425, 9475, 9630, 10150

Offset: 1

Author

Robin Garcia, Sep 23 2004

Keywords

• nonn

Comments

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.

Examples

			a(4)=10 (9^2+1)/2=41 and 10^2+1=101 and (11^2+1)/2=61 are prime.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Programs

Mathematica

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 *)

Extensions

More terms from Harvey P. Dale, Mar 19 2018