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

1,1

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.

Links

Table of n, a(n) for n=1..31.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A002522, A002496, A096012.

Sequence in context: A186509 A323085 A097548 * A344938 A261711 A128235

Adjacent sequences: A218044 A218045 A218046 * A218048 A218049 A218050

Keyword

nonn

Author

Michel Lagneau, Oct 19 2012

Extensions

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

Status

approved