A218047 Numbers n such that n^2+1, (n+2)^2+1, (n+6)^2+1, (n+10)^2+1 and (n+12)^2+1 are prime.

4, 14, 31464, 37684, 65664, 202034, 287414, 300174, 430044, 630734, 791834, 809244, 885274, 1230334, 1347834, 1411654, 1424674, 1475744, 1635134, 1721844, 1914514, 2391364, 2536414, 2855194, 3151704, 3386994, 3421844, 4010614, 4121494, 4186664, 4566484

Offset: 1

Author

Michel Lagneau, Oct 19 2012

Keywords

• nonn

Comments

This is a subsequence of A096012.

a(k)==4 mod 10 because if n==0, 2, 6 or 8 mod 10, then n^2+1 or (n+2)^2+1 is divisible by 5. When n==4 (mod 10), then (n+4)^2+1 and (n+8)^2+1 are always divisible by 5.

Examples

			4 is in the sequence because 4^2+1 = 5; 6^2+1 = 37; 10^2+1 = 101; 14^2+1 = 197 and 16^2+1 = 257 are prime.
		

Crossrefs

Cf. A002522, A002496, A096012.

Programs

Maple

with(numtheory):f:=n->n^2+1: for n from 1 to  460000 do:if type(f(n),prime) and type(f(n+2),prime) and type(f(n+6),prime) and type(f(n+10),prime) and type(f(n+12),prime) then printf(`%d, `,n):else fi:od:

Mathematica

lst={}; Do[p1=n^2+1; p2=(n+2)^2+1; p3=(n+6)^2+1; p4=(n+10)^2+1; p5=(n+12)^2+1;If[PrimeQ[p1] && PrimeQ[p2] && PrimeQ[p3] && PrimeQ[p4]&& PrimeQ[p5], AppendTo[lst, n]], {n, 0, 460000}];lst
Select[Range[457*10^4],AllTrue[(#+{0,2,6,10,12})^2+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 30 2019 *)

PARI

is_A218047(n,d=[0,2,6,10,12])=!for(i=1,#d,isprime(1+(n+d[i])^2) || return)
forstep(n=4,9e9,10,is_A218047(n) & print1(n",")) \\ M. F. Hasler, Oct 21 2012

Extensions

Given terms a(1..31) double checked by M. F. Hasler, Oct 21 2012