A181619 Numbers k such that k^2+1 = 2p,(k+1)^2+1 = 5q, (k+2)^2+1 = 10r where p, q, and r are primes.

11, 51, 61, 101, 221, 261, 571, 2271, 2821, 2871, 5071, 5651, 5761, 6561, 6951, 9751, 10461, 10851, 11621, 11711, 14961, 15911, 16551, 17171, 17601, 18511, 19071, 19551, 23151, 25261, 27351, 27751

Offset: 1

Author

Michel Lagneau, Jan 31 2011

Keywords

• nonn

Comments

a(n) == 1 (mod 10).

Examples

			a(2) = 51 because 51^2+1 = 2*1301, 52^2+1 = 5*541, 53^2+1 = 10*281.
		

Links

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

Crossrefs

Cf. A002144, A005574, A002731.

Programs

Maple

with(numtheory):for n from 1 to 30000 do : x:=n^2+1:y:=(n+1)^2+1:z:=(n+2)^2+1:x1:=factorset(x):y1:=factorset(y):z1:=factorset(z):n1:=bigomega(x):n2:=bigomega(y):n3:=bigomega(z):if
  x1[1]=2 and n1=2 and y1[1]=5 and n2 = 2 and z1[1]=2 and z1[2]=5 and n3=3 then
  printf(`%d, `, n):else fi:od:

Mathematica

ksQ[k_]:=And@@PrimeQ[{(k^2+1)/2,((k+1)^2+1)/5,((k+2)^2+1)/10}]; Select[ Range[30000],ksQ] (* Harvey P. Dale, Sep 01 2013 *)

PARI

forstep(k=1,1e5,10,if(isprime(k^2\2+1)&isprime((k+1)^2\5+1)&isprime((k+2)^2\10+1),print1(k", ")))